What you are asking for is difficult to discern so clean up our question.
It seems like you want to create a Dragonborn and Phoenix Hybrid. I think you should run this by your DM first. As they may not want that kind of homebrew. Otherwise It just seems like you want a dragonborn with Scourge Assimar abilities. Which isn't difficult really. Dragonborn don't really have alot. So you can make a subrace.
Otherwise just make A dragonborn Sorceror who's a pyromancer.
Sounds like they have the right idea, take a feature from another race that gives an effect that is Phoenix like, and then trade it for something similar off the dragon born, like losing the resistance or something in exchange for the other ability.
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"Where words fail, swords prevail. Where blood is spilled, my cup is filled" -Cartaphilus
"I have found the answer to the meaning of life. You ask me what the answer is? You already know what the answer to life is. You fear it more than the strike of a viper, the ravages of disease, the ire of a lover. The answer is always death. But death is a gentle mistress with a sweet embrace, and you owe her a debt of restitution. Life is not a gift, it is a loan."
OK... so her back story is that a mad wizard decided to fuze a dragon and phoenix egg with/into a haft human/elf female child. and just because he thought it is good added magic power crystal into her body to increase her power she can take on the full form of a dragon or phoenix with their powerful ability and also anywhere in-between. as he created her the magic went wild and exploded killing him and destroying the area around when she wakes she is an 18yr woman with a foggy memory of how she came to be there but with a full understanding of her powers which she finds strange. the first place her finds is a farm which is run by a monthly old woman who without asking any question takes her in as a daughter this farm attacks a couple of years later by monsters and the old women and farmhands are killed in the fight she sets out to protect the common people like her adopted mother. what do you think...
I mean, whatever you come up with is going to be 100% homebrew, so whatever your DM says is cool.
However, what you describe (a level 1 character that can turn into a full-blown dragon or phoenix at-will?) is WAY too powerful. Not a chance I'd allow something like that in play. Maybe as a level 20 capstone, sure, but not before that.
The easiest way to do what's been described is to just take the traits from red Dragonborn and then write your character's appearance and backstory to fit the concept you have in mind rather than the race, and then reskin abilities where appropriate to make them fit. This is true for most mixed breed characters. The main thing you'd be changing in that case would be the Breath Weapon feature, and you could easily change it to something like:
Phoenix Surge. You can use your action to ignite the elemental energy within yourself.
When you use your Phoenix Surge, each creature in the area must make a saving throw. The DC for this saving throw equals 8 + your Constitution modifier + your Proficiency Bonus. A creature takes 2d6 damage on a failed save, and half as much damage on a successful one. The damage increases to 3d6 at 6th level, 4d6 at 11th level, and 5d6 at 16th level.
(You could also use the regular draconic breath weapon stuff when you wanted if you suddenly felt like emphasizing your character's draconic features. No need to only have one or the other when they're both mechanically the same.)
Note that mechanically it's exactly the same as Breath Weapon, but now it reflects the special ability of a character with phoenix traits. You'd be hard pressed to find a DM who'd deny this, since it's just a fluff change with no mechanical benefit. You could look like a half-elf most of the time, but with a strange resistance to fire and the ability to unleash flames from your body every now and then, perhaps a few draconic traits here and there, and it'd be entirely mechanically sound since it's just appearance changes. You can achieve most exotic character concepts while exerting almost no effort by reskinning things; Imo the only time you should completely homebrew something is when there's nothing comparable to it you can reskin, and even then it's encouraged for you to steal and reskin mechanics from other things as much as possible.
Take Draconic Bloodline Sorcerer as your class and take either Folk Hero or Far Traveler (based on the backstory you provided) for your background and you'll be set.
If you want to be able to transform into a dragon or a phoenix... Well, you certainly won't be able to at will in most games; The DM might use it as a plot device here and there, but having an actual feature that says you can is something you're unlikely to get. Keep your character reasonably within the power band of the rest of the characters. If you want special abilities or concessions then that is a conversation to be had with your DM.
Also your backstory is pretty weak imo. Ditch the elements that would imply your character is super strong or has some crazy latent abilities because that stuff is almost always toxic to the game. Expand on the relationship between your character and the relationship they had with their adopted mother as well; The whole "adopted with no questions asked" part is a bit too convenient. Gonna quote Pixar's 19th rule of storytelling here: “Coincidences to get characters into trouble are great; coincidences to get them out of it are cheating.”
I went to mix Dragonborn and Phoenixheated to make a character I'd love to play as I've privately daydreamed about sort of being can you help
What you are asking for is difficult to discern so clean up our question.
It seems like you want to create a Dragonborn and Phoenix Hybrid. I think you should run this by your DM first. As they may not want that kind of homebrew. Otherwise It just seems like you want a dragonborn with Scourge Assimar abilities. Which isn't difficult really. Dragonborn don't really have alot. So you can make a subrace.
Otherwise just make A dragonborn Sorceror who's a pyromancer.
Sounds like they have the right idea, take a feature from another race that gives an effect that is Phoenix like, and then trade it for something similar off the dragon born, like losing the resistance or something in exchange for the other ability.
"Where words fail, swords prevail. Where blood is spilled, my cup is filled" -Cartaphilus
"I have found the answer to the meaning of life. You ask me what the answer is? You already know what the answer to life is. You fear it more than the strike of a viper, the ravages of disease, the ire of a lover. The answer is always death. But death is a gentle mistress with a sweet embrace, and you owe her a debt of restitution. Life is not a gift, it is a loan."
OK... so her back story is that a mad wizard decided to fuze a dragon and phoenix egg with/into a haft human/elf female child. and just because he thought it is good added magic power crystal into her body to increase her power she can take on the full form of a dragon or phoenix with their powerful ability and also anywhere in-between. as he created her the magic went wild and exploded killing him and destroying the area around when she wakes she is an 18yr woman with a foggy memory of how she came to be there but with a full understanding of her powers which she finds strange. the first place her finds is a farm which is run by a monthly old woman who without asking any question takes her in as a daughter this farm attacks a couple of years later by monsters and the old women and farmhands are killed in the fight she sets out to protect the common people like her adopted mother. what do you think...
There is a phoenix sorcerer subclass in UA which might have what your looking for.
So the character you have in mind is both a Dragon and a Phoenix that just happens to look like a half elf every once and a while?
She/Her Player and Dungeon Master
I mean, whatever you come up with is going to be 100% homebrew, so whatever your DM says is cool.
However, what you describe (a level 1 character that can turn into a full-blown dragon or phoenix at-will?) is WAY too powerful. Not a chance I'd allow something like that in play. Maybe as a level 20 capstone, sure, but not before that.
The easiest way to do what's been described is to just take the traits from red Dragonborn and then write your character's appearance and backstory to fit the concept you have in mind rather than the race, and then reskin abilities where appropriate to make them fit. This is true for most mixed breed characters. The main thing you'd be changing in that case would be the Breath Weapon feature, and you could easily change it to something like:
Phoenix Surge. You can use your action to ignite the elemental energy within yourself.
When you use your Phoenix Surge, each creature in the area must make a saving throw. The DC for this saving throw equals 8 + your Constitution modifier + your Proficiency Bonus. A creature takes 2d6 damage on a failed save, and half as much damage on a successful one. The damage increases to 3d6 at 6th level, 4d6 at 11th level, and 5d6 at 16th level.
(You could also use the regular draconic breath weapon stuff when you wanted if you suddenly felt like emphasizing your character's draconic features. No need to only have one or the other when they're both mechanically the same.)
Note that mechanically it's exactly the same as Breath Weapon, but now it reflects the special ability of a character with phoenix traits. You'd be hard pressed to find a DM who'd deny this, since it's just a fluff change with no mechanical benefit. You could look like a half-elf most of the time, but with a strange resistance to fire and the ability to unleash flames from your body every now and then, perhaps a few draconic traits here and there, and it'd be entirely mechanically sound since it's just appearance changes. You can achieve most exotic character concepts while exerting almost no effort by reskinning things; Imo the only time you should completely homebrew something is when there's nothing comparable to it you can reskin, and even then it's encouraged for you to steal and reskin mechanics from other things as much as possible.
Take Draconic Bloodline Sorcerer as your class and take either Folk Hero or Far Traveler (based on the backstory you provided) for your background and you'll be set.
If you want to be able to transform into a dragon or a phoenix... Well, you certainly won't be able to at will in most games; The DM might use it as a plot device here and there, but having an actual feature that says you can is something you're unlikely to get. Keep your character reasonably within the power band of the rest of the characters. If you want special abilities or concessions then that is a conversation to be had with your DM.
Also your backstory is pretty weak imo. Ditch the elements that would imply your character is super strong or has some crazy latent abilities because that stuff is almost always toxic to the game. Expand on the relationship between your character and the relationship they had with their adopted mother as well; The whole "adopted with no questions asked" part is a bit too convenient. Gonna quote Pixar's 19th rule of storytelling here: “Coincidences to get characters into trouble are great; coincidences to get them out of it are cheating.”