I am extremely anti affairs. I think they are cowardly and abusive.
However, I really wish that this message that lack of sex or physical intimacy is perfectly acceptable in a marriage wasnt so rolled out. The lack of it has the potential for a huge problems that leaves your relationship vulnerable to ending especially if it's at the decision of one person.
It is literally one step away from moving your relationship into co parents and roommates who co ordinate on schedules/bills and house organisation.
The level of acceptance of that on here and then is surprise that one person is going outside the relationship for sex, intimacy and a romantic relationship is pretty naive.
The only way it's anyway healthy or acceptable is if it's something that is openly discussed, agreed on and both people are happy with it.
That doesn't mean its ok for a partner to cheat, but it does mean its highly likely at least one person if not both at some level wont be happy in the relationship.
Personally I'm not a person who could sit on something like this, trying to go detective and figure out it out.
I would literally put the kids to bed, ask him to sit at the kitchen table put his bag down on it take out the condoms and ask him is he having sex with someone else.
You mention working, how are your finances? Do you have access to them all?
Have you support to call on outside of him?