I'm guessing if you're fishing with a swing head jig you'd be using a floating plastic bait and you'd want it to be of the crawfish fish type so it looks like a crawfish with it's pincers up. Right?

If that is the case what is the best bait to use with a swing head jig?

I have Ragetail Chunks, Craws, Rodents, Big Bite Baits Fighting Frogs, and others. But I think all of these sink.

Most of the baits I fish on one sink. Pretty much the only bait I don't fish on one is a tube, and I'm sure they would work too. Baby brush hogs, ribbontail worms, stickworms, finesse worms, beaver baits, boot tail swimbaits, craws, skirted twin tail grubs, lizards, pretty much anything. Very few of the baits I fish on one float. 

Everything works great. I use swing heads and Jika Rigs with any soft plastic.