I’m not sure I believe in people being literally sex addicts, but I do believe people who cheat want to find ways to excuse avoid the (deserved) consequences of betraying their partners - it would be great for him, if he could convince you to believe he has a disease, rather than just admitting he wants his cake and to eat it. You’d be likely to stick around trying to fix him. Because that’s what a caring, supportive partner would do if their spouse contracted a real disease. I might be wrong, but I think he’s playing you.
When he says “I am a piece of ****” he’s, to paraphrase from a blog about getting your life back after being cheated on, playing the Sad Sausage Channel.