My ex problem

Hi I'm new here. I need some help. To make this long story short, I've been with my now ex boyfriend for 10 years. I recently broke up with him for many many reasons. To make the situation more difficult we have 2 children together. My issue is that I'm still living with him (temporarily) and I've been "seeing other people" but I feel like I'm cheating because even though we're no longer together I'm  still living under his roof and he doesn't seem to understand that we're no together because he still calls me "his woman". What should I do?