Can an attacker steal a very recent tx and alter it, broadcasting a new tx with his address on the output? I'm reading the technical FAQs on the wiki, and what I don't see is, if I create a transaction with signed prev_outs and new outputs, do I also sign the ENTIRE new tx before broadcasting? I don't see any field in the spec for a final overall signature. 

So what I'm asking is, if I receive a tx from a peer, what is stopping me from altering that tx message so the output is now my own address, and trying to broadcast my new fraudulent tx to the network faster and to more peers than the original honest tx?
In a typical transaction, every signature in the transaction is a signature over all the outputs. So you can't change any of the outputs without invalidating all the signatures.