Can I adjust spell slots/learned spells for my homebrew caster?

That is impossible. All you can do is give it the exact same spellcasting progression as the Eldritch Knight. You cannot give it full Spellcasting, you cannot give it half casting (like Artificers, Paladins and Rangers), you cannot alter the progression of spells known or spell slots in any way. This is all you can do:

https://www.dndbeyond.com/forums/dungeons-dragons-discussion/homebrew-house-rules/119032-how-to-add-spellcasting-to-subclasses-for

It would likely be easier to make it as a Paladin subclass since they already get 5th-level spells, you wouldn’t be able to integrate the Cantrips like an Eldritch Knight, but you could fake it.

Edit: What you could do is fake 5th-level spells for that fighter the same way you would fake normal Cantrips for the Paladin. They would end up functioning like a Warlock’s Mystic Arcana.

The Eldritch Knight isn’t a “half caster,” it’s a “third caster.” DDB only implements the official stuff. The only official Spellcasting subclass feature is the one that the Eldritch Knight and Arcane Trickster use. Ipsofacto, the only ⅓ Spellcasting progression available for use in a subclass for martial classes is that one. You could change the spell list from Wizard to Cleric, Paladin, or both. You just can’t change the number of slots, casting progression, or spells known.

It is what it is. Sorry I couldn’t give you an answer you liked.