So how can I ever trust him again or believe what he says? Gently, you can't. Not now. Not when he's still in contact and hiding it/lying about it and refusing transparency.
. I don't think the affair is continuing In the aftermath of d-day, continued contact (and dishonesty about it, in particular) IS still being in the affair.
The ball's really in his court. Words are easy to say, but actions are what tell the real story.
You ask "why do I feel like this again?"
It's because his actions don't match his words.
Sadly, you have not been in reconciliation. Whether you will enter it relies on two things. You can control one: your willingness to offer the gift of R and to work with him to repair the damage he's done to the marriage.
But you can't control the other---his willingness to (a) END the affair and ALL contact, (b) learn new coping skills, (c) establish and maintain strong boundaries, (d) explore WHY he was willing to engage in behavior that harmed himself, you, your marriage, and your family, and (e) work with you repair the damage caused by his actions.
And without the latter, you can't reconcile.
Plain and simple.
I hope he figures this out before it's too late.
Sadly, there is no way for you to make him.