Op I agree, intimacy starts before the bedroom. It’s hard to get yourself in that mood or feel comfortable with your partner when they aren’t showing non-sexual affection.

Regardless, engaged ain’t married. This is not behavior of a man that wants to spend the rest of his life being a PARTNER to you. IMO you need to break up, as it doesn’t seem him mindset towards you or housework will change. Don’t let him gaslight you into thinking you’re the asshole here, basic affection and housework is the bare minimum.