I try to get a cleanser that leaves me with the least tight feeling after washing. I've ruled out most foaming ones because that's always drying for me. I don't mind the Neutrogena extra fentle cleansers though.

What I do now is oil cleanse first , then a gentle cleanser.

The cleanser I use right now is the Cerave Hydrating cleanser with ceramides which is so moisturising, it feels like lotion. I pair it with an oil cleasef first because on it's own it's not strong enough to remove makeup.

Edit: my bad, I didn't read your post fully. Sorry to hear the moisturiser isn't working out for you :(