I am wondering if I should ask H to make the move as soon and as quickly as possible
I would not. Keep acting like it doesn't effect you one way or the other - that is what detaching is all about (I think). DjV will offer better advice in this area than me but answer any questions your kids have as honestly as you can without casting blame. When they are around try to send out vibes that you are ok (hard i know). They temp check too - and they mimic the temperature you and your H set. GAL with them where you can. This might be harder for you as your kids are teens and don't really want to GAL with their parents, but maybe go out for meals, go bowling, watch a game together. Show, with actions, not words, that you are there for them.
Is there a counsellor / form teacher you can talk to at their school?