I'm your H in this situation, although didn't pester for sex, but did tell him if he wasn't interested in a sexual relationship then I would look elsewhere (note, this was after 5 years of no sex, and only had sex twice in the previous 2 years). He said go on then.
With hindsight, he had spent so long not considering me as a person at all, just as a role fulfilled within the family, that I think he really believed that no one else would find me attractive either.
They did, and I didn't (and still don't) consider I cheated on him at all. BUT, I realised, after the initial euphoria of being with someone who actually saw me as a person, that this type of relationship also wasn't what I wanted so I left. Not for the other person, that ended also, but because I'm not naturally deceitful and I was having to lie to my children about where I'd been and who I'd been with.
Tbh, I would use this weekend to openly discuss how you can both move this marriage forward, either together or apart, and be honest about how you want your life to be. You both have a right to feel valued and loved, and it doesn't sound like there's much of that around at the moment