In case there's a closer vet you've overlooked, this may help you locate them: https://members.arav.org/search/custom.asp?id=3661
Your snake really does NEED vet care- the bites from a rat are likely to get infected & may cause great harm to your snake- he should probably be on an appropriate antibiotic from a herp vet.
It's never "fine" to leave a live rodent (especially a rat!) in with a snake. Feeding pre-killed rodents is much safer for pet snakes- because sooner or later, your snake will be injured by live.
I agree with the post above by Homebody for first aid treatment of your snake's wounds, but it won't stop a systemic problem from the rat bites. And the behavior (unsteadiness) in your snake sounds like he may be ill anyway- it was a bad idea to feed him when he's in this condition.
When snakes bite themselves, it can be a response to internal pain they are feeling- as if they think they're defending themselves from whatever is causing that pain. If you cannot afford the time or money to take proper care of this snake (especially right now-he needs immediate help!) then please give him to someone who can & will provide for his needs. We owe our pets that much.