"We've been together nearly 25 years and to throw it away feels wrong without trying"
Do not get stuck on the sunken costs fallacy, this is what your above comment really is. This causes you to make poor relationship decisions. He did not try and chose to embark on an affair instead; no-one forced him to do this.
People get bogged down by focusing on their sunk costs.
There are two ways to understand this process, both involving avoidance. One is an avoidance of disappointment or loss when something doesn’t work out. When a relationship doesn’t succeed, especially after a long period, especially after many shared experiences and especially after developing a hope that the relationship would be a good one, it is a loss. It is a loss of what might have been and an acknowledgement that a part of one’s life has been devoted to this endeavour.
Another angle to evaluate is that focus on “sunk cost” creates a distraction from one’s inner truth. The sentence often goes like, “I’ve already invested too much, so I can’t notice my thoughts and feelings that are telling me to end or change this relationship.”
This is a type of insidious defense against noticing yourself. You enter into a neglectful relationship with yourself which divorces you from your inner thoughts and the quiet feelings that might guide you in your life. In other words, thinking about what already has been may prevent you from deciding what you want your life to be.
The key is to clear away the distractions to rational and emotional clarity. Getting stuck in your “sunk cost” prevents you from this clarity.
No-one forced him to have an affair; that is all on him. He felt entitled to do that. He never really tried to resolve any problems within this marriage. At the very least he should give you time and space apart from him, he should be living elsewhere now particularly if you and he are still under the same roof.
What else has he done here to show you how supposedly contrite he is?. I would also consider having counselling for your own self. Have a read too of "Not just friends" written by Shirley Glass.
What do you want to teach your children about relationships and what are they learning here?. These people cannot and must not be used as the glue here to bind you and he together. Staying for the sake of the children rarely is a good idea because it also teaches them that your relationship with their dad was based on a lie. Its a terribly heavy burden to place upon them and one they will not thank you for doing either. Whose sake would you be staying for; theirs or perhaps more to the point yours?.