My ex did this. Very similar situation, very similar excuses. He had also claimed low libido for the reason we were barely having sex, then after a year or so of this I found out he was on multiple hookup sites searching for women around our home and local to somewhere he visited regularly, and he'd been messaging sex workers (while I was in the house, too). Like you I had a feeling something was off but he was covering tracks; I ended up finding out because I eventually saw concrete evidence that he was definitely lying about something and signed into his devices to get the truth. Huge shock to see all of that, not what I was expecting!
I wouldn't usually give advice like this on someone else's relationship, but on this occasion I'm going to say run.
It sounds like he’s trying to make you feel guilty for causing him to behave like this, like his behaviour is somehow your responsibility. It isn’t. I was also led to believe I was being ‘controlling’ - looking back now I know I wasn’t being controlling at all, his behaviour was overtly shady and I was reacting to that. Accusations of me being ‘controlling’ was just him being annoyed that I was asking questions about his suspicious behaviour, meaning he had to come up with lies, which he didn’t like. How dare I wonder what he’s doing on his phone when he’s acting like he’s up to something because he's messaging sex workers, what a controlling bitch!
I can totally understand how you could believe that you’re somehow responsible, but that really is just his manipulative bullshit - this is on him and him alone. My ex also tried to claim he was trying to see if he could get his libido sorted out, as though he had been sending revolting messages to other women with our relationship in mind somehow. What a prince!
Like yours, my ex also claimed he’d never gone through with it. I’ll never know the truth on that one, but I will say that I stayed with him and he ended up cheating on me and leaving. He was shopping around, whether he acted on it at that time or not. Do you think you can ever trust your partner again after this? Do you want to be in a relationship where your self esteem is taking a battering, you’re having no sex and you’re always wondering if he’s trying to get it elsewhere?
I’m so sorry you’re going through this. Talk to friends in real life, tell them what’s going on and get some support. You are in no way overreacting, this is a genuinely horrible situation and it isn’t your fault in any way.