Is my husband cheating?

I keep finding text messages, emails and secret messaging apps on my husband's phone to other women about meeting up and having sex. Even to escorts to meet at motels. They're all made while I'm at work, he's at work or when I'm sleep next to him. He says he's just catfishing cause he thinks it's funny but he gives out his real phone number. email and sends nudes of himself. Since I only have the messages he says since I didn't actually see him having sex with another woman then he's not cheating. or that I'm making it up and copies of the messages aren't proof and I'm a liar. Since I did not catch him in the act then I have nothing. I know I sound stupid but any advice would help me out. We are still married and together but I have my doubts... should I stay or leave without seeing it for myself?
Is my husband cheating?
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