I don't know that it is necessarily best to "be still" unless you can live with the way he is acting right now.
If you can detach from it and are "OK" with things then do what you think is best for you. If that is to be still and go with the flow, then OK.
I you can't detach from it, and you are miserable wondering if he is seeing her, if he is sleeping with her, knowing she's still in his apartment, knowing he's lying to you by saying he's going to fix it then doing nothing then i don't think being still is the best thing.
I don't know that always doubting and questioning is the right thing either.
It's kind of like death to your marriage by a thousand cuts.
Basically you have choice to make and only you an decide what is best for you.
1. be still and hope he starts acting like a real husband soon.
2. Take some big and drastic action to let him know you will not tolerate his behavior any longer.
I would suggest to pray about it. I liken your situation right now to pre my first d-day. I knew deep down that he was cheating on me. I knew when I found out that I was going to have to do something - even if that something was NOTHING. I knew I needed a game plan for when it happened.
I Prayed for weeks before looking for evidence. Weeks of pure hell knowing he was seeing her the entire time, but when the time came I had a game plan and was able to carry it out even though it was scary as h$ll.
Praying for you and your H.
BFM
There is only one person who could ever make you happy, and that person is you. David Burns, Intimate Connections