You sound anxious. And that is understandable. You have a lot on your plate. Cheating aside, I think your dh is massively underestimating the impact a young baby and small child is having on your life.
My dh isn’t good at telling me he’s going to be late. Despite repeated requests and arguments to call me by x time to give me an eta. It’s not a control issue for him. He’s too engrossed. I end up calling him. So I do understand, including the cheating.
Could you perhaps call your dh every day with your kids just to say hi to daddy and get an eta? I’m trying to suggest you take a bit of control but not in a controlling way. It will give you a) a few minutes relief and something to talk about with your eldest and b) a way to reduce your anxiety about what he is doing with his time.
The way you are both communicating isn’t working and you both need to change that dynamic. Say for example he has no intention of cheating ever again, should he have to prove over and over again to you he isn’t cheating? Should the fact that he has cheated be a life sentence? Personally I think the answer to both questions is no. Because if he’s not being sneaky at all, you’re more likely to drive him away.
If otoh he is giving you reason to believe he is cheating and sneaking around, you are worth so much more.
One of dh’s colleagues had a wife, who was on his case even if he was a few minutes late. She was struggling with little kids and it was very sad for her. She moved countries for his job away from her support structure. He complained to dh and his colleagues about her after they split and she was painted as a bit crazy. I met up with her a couple of times while they were still together, she was a mental wreck and regretted him leaving the military, where he came home at a specific time. Poor woman. The circumstances were too much for both of them. He did move into a relationship with someone else very quickly and it was an emotional if not physical affair imo. Afterall this ow was easy to get On with. No emotional baggage.