I'm making a white dragonborn character, but I want to do the sorcerer class. I don't own a dnd manual on dnd beyond, which leaves me with the basic draconic bloodline option. I want your input on the choice I want to make: I want to choose the silver dragon ancestry for the draconic bloodline. What I'm thinking is that the spirits of the white dragon and the silver dragon are both inside of his head, and they constantly order him to do things, as well as argue with each other. If I had any dnd manuals for dnd beyond, I would multiclass into the barbarian path of the ancestor.

What do you think?

The short answer is that you can do whatever you want with that as long as it is in line with what you and your DM set out as far as worldbuilding is concerned.

The longer answer is that as far as I've read into it, a white dragonborn draconic sorcerer is not likely to have a "spirit of a white dragon" inside their head anymore than an elf is going to have some greater fey spirit embedded in theirs.  The type of dragonborn you are really is just about the color of your scales and element of your breath weapon.  They are quite far removed from the dragons they came from (and "came from" is an open question as to how).  Nor are they likely to have a silver dragon spirit, as that's starting to sound a bit more like a warlock than a sorcerer.  It could be that you have both white and silver dragon ancestry, while one manifests as your dragonborn type and the other manifests as your sorcerous powers, and you could even have decedents of those original dragons be living and have an interest in your character...

I like that! The character could have a white dragon "aunt" and a silver dragon "uncle" that just want to see him. Knowing my DM, he would turn it into an awkward they-arrive-at-the-same-time kind of moment.

Actually probably not as they would be far too likely to fight to the death on the spot - they are the same sort of foes as gold and red dragons. But as Sabin says it’s not about the voice n your head, it’s about the dna in your blood. Given that both have a cold breath weapon you might actually have some silver scales mixed with the white as one proved dominant in some places and the other in other places.

I think that the flavor of having two conflicting spirits in your head is amazing for roleplay, and it should be fine to go with that even if that's not the official way dragonborn work. After all, players are meant to be out of the ordinary! If you do decide to go with being descended from both, that isn't a issue either! One of the most prominent dragons in the Forgotten Realms, Imvaernarho, is a Gold dragon red dragon hybrid, which is what gives it such amazing powers. Awesome roleplaying opportunities are almost always fine and welcomed by DM's, even if it's a little out of the ordinary. Some alternate ideas could include some connection to the ancient dragon primordial Io (Not Ao, don't get them confused lol). He was the dragon of balance, with Tiamat and Bahamut as his children. Perhaps now, years after his death, after chromatic and metallic dragons have fought for aeons, he has called upon the souls of two dragons that are powerful enemies to work together at last in a "human" form, to finally bring balance to dragonkind.

Just ideas, and feel free to play whatever sounds most fun to you!