Welcome to PC aliveindeed. Usually if women aren't educated about cross dressing, they might worry their man is gay - is this a concern for you? To reassure you.....the majority of cross dressing men are straight. Its up to you if you can accept him being discreet and continue your relationship. For some its a deal breaker but some women have even tolerated the partner doing it around them. Men who do this usually knew in their teens they like this, so insisting him to stop may not be worth the trouble.

The bigger problem is the sexting and him talking about you when you're fighting. Maybe consider couples counseling. Both of you need to solve this problem and figure out if you have a future. Best of luck.