The disloyalty is appalling. Your DH talking to another woman and telling her your relationship is going wrong, and he may want to sleep on her sofa? He's fishing, OP - seeing if she is up for it with him. Which doesn't mean an ongoing affair, could be purely sexual but thats bad enough. She's no good either - repeating aspects of a private conversation between you, and putting you in a bad light. & the responding to his compliments, talking about going on a diet etc its the age old game. Its the disloyalty that would put me off both of them. Your DH is worse but in your shoes I don't think Id be splitting hairs about that, Id want it sorted out asap
If you don't want to end your marriage then yes, go on a couple of nights out with your DH, get nice and glammed up for them too, helps with confidence. & if he brings her up during your night out then don't mess about - be direct and tell him you feel he and his cousin talk too much, and too often, and you aren't comfortable with it at all. Don't worry about seeming too forward, just tell him.
You don't have to repress your feelings for anybody, as women we are taught far often to do that, go with the over-softly softly approach when someone has done wrong to us. & that if we respond to that wrong, we are somehow worsening the situation or pushing for an ultimatum. I don't buy any of that, getting things clearly out in the open is best. You are an equal in the relationship.
& goes without saying - no private conversations with her in the future, if she brings up anything or tries to get personal then shut her down immediately, non-committal responses. In fact in your shoes Id do my best to have as little to do with her as possible
The 'phone thing is difficult - I know its deemed wrong to look but if you have suspicions I absolutely think its the right thing to do. Mobile phones are a Godsend to cheats. If you didn't look you'd be none the wiser re their shitty disloyalty, and would go into a discussion with him without knowledge of the type of conversations he's been having with this woman. As unpleasant as it is, its best to know so that you aren't wrongfooted. If he left to sleep on her sofa I wouldn't touch him again with a bargepole if I were you, but you know best how you would feel about that.
Hopefully he will see sense and leave this woman alone. Good luck