These are great examples of ellipsis:
1. I was born (a) wild (child).
2. I was born to be (a) wild (child).
(Note that, a non-ellipsis reading admits a difference in meaning:
As for this structure,
1. I was born wild. Cf. I was born happy.)
2. I was born to be wild. Cf. I was born to be happy.)
3. Dogs run free.
if free could in fact function adjectivally, the meaning would be free dogs run, a different meaning from dogs run free.
The simple answer is this. Free is short of freely, an adverb.