"" I don't think an unsigned contract will hold up too well.""
I do think that the partial payment ( if it was by check) is as good as if he signed the contract. Its his agreement (intent) to have you do the work and pay for it.
When I have trouble with customers paying the final payment, I remind them of the "Blackboard" at all the areas supply houses. They say "whats that ?" I tell them their name & address is posted as (did not pay for services) for all for all the companies they may try to call for service.