That text could indicate he's found one but didn't use it. Or went in there but didn't personally pay for a hand job. Yuck. But regardless, I actually think you should consider ending things. He'll lie anyway about it and can you ever trust him?
You sound deeply unhappy with him. After my marriage ended, I became a better parent, I was more at peace, more stable for my 2 DCs and I actually get on better with ExH now we do not live together. Once the dust settled and we found a good balance on access arrangements etc, we now get on so much better. I'm happy, kids happy. And I sleep well now, instead of worrying where he is, what he is up to and if he is telling me the truth.
It's a huge step and terrifying but ask yourself this? What does he bring to you? How different would things be without him? Would you be happier?
He sounds like a real charmer, a good catch not
He can still be a good Dad, if he is one, and he can still take his child off to places and do special things with them, without you being together, but you can move on with your life and be happier. And perhaps that would be better for your child, if they're picking up on hostilities/ arguments etc too.
I hope you have a good day wherever you go. I suspect your mind is like a washing machine right now though. Everything is clearer in retrospect. I do not often say this but LTB.