I think the basic idea has potential, but as
@Remathilis points out, there are a lot of pitfalls here. The class is thematically built around an "ultimate form," but most of its actual power is in the spells you have to cast to reach that form. The ultimate form itself is more a consolation prize for running out of arcane firepower, and that isn't going to make it particularly fun to play.
If I were designing a take on this class, I'd make it a half caster and give it a set of "ultimate form abilities," something like:
Dragon Scales. At level 1, when you cast a spell of 1st level or higher, you can use a bonus action to gain dragon scales for 1 minute. If you are not wearing armor, your base AC becomes 14 + your Dexterity modifier. You also gain resistance to your draconic damage type.
Dragon Fangs. At level 5, when you cast a spell of 2nd level or higher, you can use a bonus action to grow fangs for 1 minute. Your Unarmed Strikes now deal 2d8 piercing damage and use your Charisma modifier in place of Strength. [The class also grants Extra Attack at this level.]
Dragon Wings. At level 9, when you cast a spell of 3rd level or higher and use Dragon Scales, you sprout wings for 1 minute. You gain a fly speed of 60 feet.
Dragon Breath. At level 13, when you cast a spell of 4th level or higher and use Dragon Fangs, you gain a breath weapon for 1 minute. Once on each of your turns, you can use your breath weapon in place of an attack, dealing 4d8 damage of your draconic type in a 30-foot cone. Each creature in the area can make a Dexterity saving throw against your spell save DC for half damage.
Dragon Ascendance. At level 17, when you cast a spell of 5th level or higher and use either Dragon Scales or Dragon Fangs, if the other ability is already active, you take dragon form for 1 minute. You become Large-sized and gain 50 temporary hit points, and your breath weapon becomes a 60-foot cone. Your equipment melds into your new form, but any magic items you have attuned continue to function normally.
This would obviously need a bunch of polishing, testing, balancing, and tweaking, but the basic idea is to spend your first two rounds casting spells to power up. Then you're a melee monster -- literally -- for the rest of the fight.
But that's not a good replacement for the sorcerer, which has "full caster" as a significant part of its identity. It would need to be its own thing. A full-caster version might give the ultimate form less baseline power, but provide ways to burn spell slots in ultimate form for extra punch -- e.g., each use of your breath weapon costs a spell slot and does damage based on the slot level.