FWIW, I think you are now poisoning your M. You have made big progress. Holding this secret in is going to taint every single interaction you have with H. If you want something to send him running, continue to live with him while holding this in.
The money stuff is an excuse. His financial responsibilities will not change immediately if he suddenly moves out. It will probably save you both money if you file jointly, whether he is living with you or not.
Right now, you have no idea what is going on with H and OW. How about some honesty and directness.
"H, I was surprised to see a picture of you and OW skiing on your new phone. I had assumed you had cut off non-business contact with her, but maybe that is not the case. I need to understand what the status is of your interaction with OW as it affects my choices and how I want to live my life."
You also need to know what you want: -- 0 non-business contact -- her number blocked on his cell phone -- whatever
