How much privacy is needed in a relationship?

My boyfriend and I moved into a new place together. I was trying to get on my email on his computer since his was set up and had the WiFi set up and all that good stuff. I had to sign him out of his account before I did mine but I saw some convos with some girls names i thought was unusual for email. I asked him about it. We talked, no big deal. Fast forward to three weeks later, I’m signing into my email on his computer again but this time he’s blocked all his shit. I can’t get on his email or anything. I hate to feel like I’m being nosy but it wasn’t an issue before and now suddenly it’s a problem. I can’t just ask for his password, he would NEVER go for that. I’ve tried before with his phone.

What are y’all thoughts? Am I overthinking shit? Is he hiding something?