I agree about Wobbly red flags, namely he has no friends, he is ashamed, he feels the need to deceive in order to get an ego boost.
Those are really deep set issues which it would take a lot of hard and sustained work to address.
I also agree that the change would be better if he were not present in your home, for numerous reasons mentioned by others.
On the other hand, I don?t think it is true that people cannot change sufficiently, and I do think that getting caught can act as a catalyst. Clearly, its better for someone to come to realisation by other means, but many of us in different ways come to change from enormous catalytic shocks of some sort or another.
My experience of an H who was sex chatting OW online and exchanging photos for some years is that it has acted as a catalyst. He stopped 21 months ago.
However, but........
It is much harder than the philanderer realises
Many of the changes he needs to make are not about sex, but about his whole attitude to life, you, others.
That is a great deal of work for the philanderer
It is inevitably diverting for the betrayed spouse, even when you try for it not to be, and I assume even tougher if the children are younger.
You would have to be prepared to police yourself not to police him- the urge to check and snoop etc etc if very strong, I found, even if that is not natural to you
Every small failure takes you back to the beginning.
I agree totally with Wobbly that you must resist every urge to look at his ?whys? and act as his support and counsellor. I didn?t do that well at all at the start, for months, and thoroughly regret it. It is simply not your problem, issue or business, and you can never be his mum or therapist, even if you were beforehand to some extent.
I agree that you need to focus solely on what you want to do next, and also give yourself time to do that. You don?t need to make a decision now, and in some ways it may not be good to, as you are too stunned, let alone distracted by a new baby.
I think I would ask him to set himself up elsewhere, leave the door open whilst you think, and let the next steps emerge.
The last point I would make is that you have not been together for so long, it seems. That would possibly have made a difference if it were me. As it happened we were 15 years together before the infidelity started and 21 before I discovered it, and with teenagers who had opinions to contribute. Even then, we moved house and partly as I bore in mind that leaving him at a later date, if necessary, would be easier to do with a smaller mortgage.