Help me understand...Please 😞

Okay, so I'm demisexual meaning I'm ONLY attracted to people I have an emotional connection with. When my husband & I first started dating and I told him, he said he's the same way. But he's not. Multiple times I have caught him messaging other women, just flirting but it still really hurts me. He breaks down and apologizes, says it won't happen again, he hates himself, etc. Last time he said he'd go to therapy but has only been able to make it to one appointment due to work & money. We got married in December & have been trying for a baby since. Last weekend I discovered he'd done it once again. I finally got him to admit he IS attracted to other people, and asked him why he lied. He said he didn't want to be that way so he tried to repress it. I apologized for making him feel guilty this whole time as I know it's normal for most people, and told him that repressing those feelings only leads to secrecy. We are both going to go to counseling together if possible, but I guess I just need advice in the meantime. I can't wrap my head around it & it really hurts me when I think about the fact that he lusts after other people. Please be kind, I just need reassurance that it's normal & not harmful. I'm blindsided by it every time. I do feel partially responsible because I really did make him feel like shit. The first time he tried to tell me he feels attracted to others I immediately responded that I was going to leave him. He's been repressing it ever since. Thanks in advance for any advice, and again, please be kind. This is very hard for me to accept.
*Shiona, we both know messaging other women isn't okay, which is why we want to seek counseling. He knows he has issues but I feel like it wouldn't have happened if I hadn't made him feel bad for being a normal guy.