Greetings pjlenn,
Your sketch is a bit confusing, but to answer your question, yes you will need to add the neutral to one side of your contractor coil. Use the controller to energize the coil of the contractor.
Also, it appears you are using a two pole contractor, but I can't see why you would need it unless your ac uniit is 220 volt. In which case, the diagram and proposed wiring scheme is not correct for that application.
Also, I would recommend using two duplex recepticals as opposed to a single. I have never been a fan of using the break away tab on a receptacle particularly if the receptacle is being used for two loads with different amp draws.
Hope this helps!
Good Luck!!!