I don't think it should stack, more as mesh together. They have slightly different abilities.
When you start progressing in a PrC you stop progressing in your primary class.
Edit: Unless the PrC explicitly suggests otherwise.
Edit2: PrCs are essentially templates. The DMG encourages DMs to customize the classes for their campaign setting. If a DM wants to reinvent a PrC that uses the Dragon Discipline as a bases that has abilities that stack then maybe it makes sense. Use caution though.
You know I never liked the Dragon Disiple as is. It always seemed to me that both the sorcerer (my fav.) and all its prestige classes (in Wizards offical books) were made for multi-classing and not for its own sake. I mean a Fighter for example who wants to cast spells but no intrest in all the diversity of a Wizard takes the Sorcerer. This I think was a mistake in desinge. Example the Disiple gains a few extra spells per day to his list, but at the cost of advanceing in caster levels. Great if you're Fighter who only needs mage armor or magic missle in battle, but would a Sorcere really want that? Anyways long an short I've house ruled that gains a caster level every even level as a Dragon Disiple.
Maybe the PrC wasn't meant for you. I had a player who loved dragons, and when she saw the that it allowed her to become draconic, she took it, it didn't matter what it did to her spellcasting.
Don't miss understand, loved the concept and use it often (its the PrC of choice for me and my sorcerers), but mechanically it I see as bad for spellcasting, which is one of the core aspects of the the class. Its not unplayable, simply slightly better with my modifcations.
ttfn dre