Is my (29f) (29m) partner who’s struggling mentally using unhealthy escapes or is he playing in my face/ leading me on?

I know it’s a bit long but please drop your thoughts ( lmk your gender in comment if you’re comfortable with it)

I (29f) caught my (29m) partner cheating. Things were bad for us and had been planning to get into therapy. We mostly argue because I’m a nag and he gets nagged. I do everything for our child, him, myself and our home and he works. He’s been taking off work a lot so I expected him to help with our son if he’s not working half his regular hours. I do literally everything for everyone nonstop with no break or time for myself. He has been struggling with mental health for at least since our son was born 1.5 year ago. He’s gotten on medication, but hasn’t gotten into therapy to really get to the root of the problem. He took a leave from work due to mental health. I’ve been tired of asking (nagging) him to contribute (take trash out, dishes once a week, occasionally do stuff for our son. We got into a fight and he packed his stuff and went to his families 4 hrs away. Within the first day of him being there he said he wants to work on our family and that we’re still together, but just needs space. I asked him about a time frame he refused to give me one. I’m a sahm and said if he’s going to be gone a while he needs to find a way to make sure bills are covered. He says he’ll get it figured out. I understand he needs space but this is real life and he can’t just play things by day when he has a kid and partner who relies on him financially. I just noticed he kept being hot and cold anytime serious conversation came up and wanting to text random small talk/ try to sext and express how sexually frustrated he is. So I made a fake account (not cool ik) he added the account back and started getting to know her / flirting and as life conversations came up he said he’s completely done with his kids mom (me) and no chance they’re getting back together, was telling her all our personal business and making it clear he’s been detached for a while just finally found his out/ got the courage to leave… while actively texting me. He had told me to text his family member to ask about borrowing money to get me by until he’s back because he didn’t know how to go about it, so I did. Then he blew up saying he never told me to do that. He got really mean towards me when I had the girl (me) say I think you have a lot going on and should focus on your family not try getting to know me. He was so cold and was so nasty towards me and said he was done xyz. Then after the account confirmed she’s doesn’t wanna talk anymore he texted 20 minutes later me saying he wasn’t done just mad. I just don’t know what to do. Like he’s clearly lying /cheated (technically) and I can’t bring it up or he’ll blow up and turn it back on me for making the account. Him family member is making the communication through him so I don’t know if they’re going to help or if I’m going to lose everything because our place is in my name and any job I get in our area will barley cover bills and also 1/2 our monthly bill cost is what childcare cost. Idk I’m just messed up over the fact that he’s telling me one thing and the fake account another thing and his actions of being hot and cold in my mind are lining up more with the fake account conversation. Like he was seemingly willing to end things with me and screw over his son and I for a fake girl he started texting 5 hours prior. Please don’t be hateful I’m dealing with a lot and not a complete dodo… but the financial strain on top of trying to be considerate of the stuff he’s been dealing with is definitely blocking my ability properly process/ understand. Advice/ input on the fake account thing would be appreciated. Does this seem like someone who’s struggling with mental health and just using unhealthy escapes or is he playing in my face willing to completely fuck us over.