What we (both!), did was specific to our situation but I'm happy to share.
I was quite tough on myself. Did a lot of hard thinking, a lot of reading - I had plenty of issues of my own and I wanted to make sure I was making an informed decision.
Had a series of very difficult conversations with H. Laid it completely on the line, had examples for each of my points, kept coming back to the fact that I was prepared to leave if things didn't change/improve as there was no way our relationship was sustainable in the long term otherwise.
Got him to think about his position, his motivations, had more hard conversations where he talked to me about the part he felt I had to play in our problems (that was a horrible conversation but very valuable, much more than the 'litany of faults' we'd been throwing at each other during arguments previously).
He was outraged that I'd consider 'ruining the kids lives' - I asked if he wanted to stay married to someone who felt like he now knew I did until our children had left uni, and how he saw that working - would we have seperate rooms? An open marriage? What was he really asking me to do? What did he want me to do in that scenario?
He didn't 'agree' with me but he acknowledged I was serious. We talked a lot - god we talked all the time, it was exhausting.
H agreed to stop the groping (that really upset me, more than the sulking). And he did. There were a couple of occasions when he went to grope me sort of out of habit, and he checked himelf and apologised.
He asked to be put on a work project which meant he was abroad for 2 weeks. That was good, we talked every night, he hated being away from us and it was valuable for me to realise I could 'cope' (albeit in an artificial way), as an effectively single parent.
I went away with my best friend for 5 days, H took the time off work to look after the DCs.
TBH even after all that we still argued, resented each other, but then it was like the sun was slowly coming out from behind the clouds. I had a health issue and H was amazing. He has arranged his hours so he takes the DCs to school in the mornings (this was always a horrible fight for me, often had to wake up youngest DC). He is just nicer to me, and nicer to be around.
He also started talking about stress at work, stress about our finances. We talked seriously about them (and about going bankrupt), and came to terms with that, which meant we were less angry with each other about that too.
I've gone on for pages and pages now. But we didn't have one big problem which we could solve easily, so it's hard to pinpoint what we 'did'.
The main thing for me was having a realistic exit plan, and agreeing with myself that I was going to leave if things continued as they were. Giving myself 'permission' to do all the things I felt I couldn't do ('break up the family', 'end up in a council flat', 'take the DCs out of school', 'be alone forever'), was very freeing.