How can I prove I had verbal permission to access someone's email account and website?

I had a professional relationship with someone and it went sour. For months we accessed each others email accounts and websites. We gave each other our pwords and full permission to access these accounts as our relationship was in good standing and we trusted each other. Now he is acting so crazy that he is threatening to tell the police that I had no authorization to access his email and website. I'm really afraid because I can't prove he gave me verbal permission. I did a little digging on the internet and I think he has to have my mac address to prove I accessed his account. I'm thinking of just selling my laptop since the mac address is part of the computer. So, if I get rid of my computer will I get this psycho off of my back? Also, can just the IP address give my exact location away? What is the difference between MAC address and IP address? Note: I use an unsecured wireless network for free that apparently a neighbor nearby has so I don't have any internet service or router and such. Thanks.