It sounds like your H is ashamed and not sure how to handle himself. Whether he is or isn't still with the OW, I don't know if calling him out will help at all. I imagine it might cause him to further push away in his shame. If this was the first he has reached out to you in awhile, I think your best bet would be to agree with him, tell him you believe it would be best for both of you.
Two things about that. One, his decision, he's in control, he may need to feel like he's in control to move anywhere near R. Two, he's not expecting you to agree at all. Especially, living with his folks, I'm sure they have told him how depressed and distraught you are. It will throw him for a loop and he may start wondering what you have going on.
Then take that time, once you get your own place, to really take care of yourself. I wonder if once you appear to have moved on and he's not ale to get updates from his folks...if things might change? Maybe, maybe not. I wish I had a solid answer for you.
I caught mine in a lie yesterday. Haven't called him on it because I don't think it will do any good, but it ripped me apart when he did, and I knew he did. I can't expect him to be at the same place I am, but I had been priming myself to be open to trusting him again. Today, I don't know if I can.