Yes, an oral contract is just as binding as a written contract.
Yes, it is your word against his.
However, for it to be your word against his you have to take him to small claims court, you have to tell the truth and he has to lie. It's one thing to stiff someone with the bill, but another thing to lie under oath in court. If you drag him down to court and put him on the defensive, he'll probably come clean.
When I was in college, I signed a lease from May 1 to April 30. A couple weeks before I moved in, my landlord asked me if we could make it May 15 to May 15 of the next year, because the previous renter wanted another couple weeks. I said sure, I'll just pay the rent on the 15th of every month instead of the first. That fall my landlord had a stroke and sold the property to one of the local college town slum lords. They went by the signed lease and told me if I wanted to stay till the 15th I'd have to pay another half month's rent. What could I do? I moved out on the 1st. I lost two weeks rent but I learned a valuable lesson.