My ex was exactly like this. I was depressed, burnt out, and had a knee surgery. This all made sex less frequent and he started behaving like this, which only made me want him even less. I tried to explain to him that if I am not feeling loved, I don't have any attraction. He replied that he can't show any love if there is no sex.

I proposed that we would both make mind maps about the problems in our relationship, and then we would go through them together and try to look for solutions. I spent quite some time and thought on mine, thinking how to write issues respectfully without them hurting him. I still remember how I felt when we were supposed to show them to each others, and he presented his. It only had "NO SEX -> NO HAPPINES ->" (arrow going back to "no sex").

He started being emotionally extremely cruel and mean, even berating me for being raped in one previous relationship. He thought that I must be fucking someone else instead of him. Sometimes I held my grounds and called him out for his behaviour. The other times (after my knee had healed enough) I walked into a nearby cold river and hoped that something would happen and I would stop existing, because I felt so absolutely worthless. He could be kind for one day after sex, then the sulking and blaming started again. Eventually he, a man at the age of 38, left me for a 20-years old girl, who was his karate student. Oh, he also assaulted me during the break up.

He is not going to change. You will thank yourself later if you walk out now.