Feel stupid asking this but how do you actually separate??

[runningLou]

Not to drip feed, 4 years ago I had an affair. DH found out, allowed me to stay and we have been together for the DC more or less since then. During that time we have moved house to be nearer my family, changed jobs etc. Both of us have also suffered MH issues e.g. anxiety, stress too.
I am now finding myself feeling trapped as DH seems to suspect my every move. I know this is completely my fault due to what I did in the past but I don't know what to do about it. I've recently lost some weight and met some new friends in a running club so have been messaging them. DH is extremely anxious about this. I know it's understandable. I just feel like there is no trust there.
I would like to separate but realistically have no idea how it would work out. We have 2 DC, 6 and 10. I would like them to stay in the family home if possible. Since we had the DC I have worked part-time (previously higher earner) and done all child care. I would like to keep doing this. The house is in both our names.
I earn so little that I don't know if I could move out ... I suppose I could go to my DM's but I don't really want to involve her in my mess. If I moved out would I have to pay DH maintenance?!
Is there any way one of us could stay in the family home with the kids and somehow have a shared custody arrangement where the kids were always in the same place?
Feel completely lost and bewildered at the moment but after another very difficult 6am conversation with DH in which he accused me of being 'addicted' to going running, and 'hiding' to check my phone, I feel like things are coming to a head.