You can be affected by multiple morph spells at once,
but if you morph the same body part more than once, the
second morph effect attempts to counteract the first (in the
same manner as two polymorph effects, described below).
Your morph effects might also end if you are polymorphed
and the polymorph effect invalidates or overrides your morph
effect. For instance, a morph that gave you wings would be
dismissed if you polymorphed into a form that had wings of
its own (though if your new form lacked wings, you’d keep
the wings from your morph). The GM determines which
morph effects can be used together and which can’t.
These effects transform the target into a new form. A target
can’t be under the effect of more than one polymorph effect
at a time. If it comes under the effect of a second polymorph
effect, the second polymorph effect attempts to counteract
the first. If it succeeds, it takes effect, and if it fails, the
spell has no effect on that target. Any Strikes specifically
granted by a polymorph effect are magical. Unless otherwise
stated, polymorph spells don’t allow the target to take on the
appearance of a specific individual creature, but rather just a
generic creature of a general type or ancestry.
With this in mind, I would say that the Dragon Claws spell would be dismissed when you cast Dragon Form so long as your battle form has claws. I believe all of the dragon types in dragon form do have claw attacks specific to them, so they would indeed override your Dragon Claws spell.