I am so very sorry for your betrayal. It had to be a terrible holiday period, and no doubt you were and are in shock. Be kind to yourself.
It sounds like perhaps your husband is coming out of the fog and beginning to be honest and respectful of your rightfully fragile state. You don't have to make any rash decisions. Just watch, wait, feel and trust your gut.
Do you have full access to 10 months of phone and credit card records? If I followed your post, they "flirted" for 10 months, but only got together one time around Christmas and he stayed fully clothed (??!). Maybe, but Hmmmm . . .
Since he is trying to be honest and open, I would ask him to sit down with you tonight (or soon) and together go over all the credit card statements, his email and texts, and his cell phone records. Plan your review session, work time and child care appropriately so you won't be interrupted. Don't give him a head's up, so he can alter things.
If he truly is telling the truth and coming out of affair fog, he will gladly go over all of this with you, and make sure you have complete access any time you want it. If he has employment credit cards and/or email/ cell records, he can ask for copies knowing that he has nothing to hide. If he is sincere in his efforts to show he is sorry, he will do the legwork without complaint. It is exhausting if you have to try to research it all yourself without his cooperation. I know from experience!
The worst possible outcome here is that everything is not disclosed and you are getting trickle truth. You will be able to tell by his mannerisms and thoroughness in disclosing.
You have been traumatized. Let him make dinner, wash the dishes, vacuum the rugs, clean the toilets, put the baby to bed, AND prove to you he has disclosed everything.
Prayers for healing.