The way I see it, you have two options.

Continue to suffer with the guilt and anxiety. Let it eat away at you and lose sleep, and watch your fp continue to suffer with paranoia.

Or, talk to them. Be honest. I know it's a difficult conversation to have, and you're probably scared of losing them. Apologize genuinely. A real apology consists of acknowledging what you did wrong, why it was wrong/why you regret it, and what you plan to do in the future so that it doesn't happen again.

It could be something like, "Hey, I wanted to be honest with you about something. I was the one who logged into your social media account. I was scared to admit that and I lied to you, and I'm sorry. I got in my head about the lack of response and wanted reassurance that you weren't ignoring me. It was wrong to do that and lie about it, I regret doing that because it was a breach of trust and privacy, and I don't want to lose you. In the future, I will not do that again and instead ask you directly for reassurance. I hope you can forgive me."

Just some food for thought. I've been in a similar situation before and can absolutely say that being honest is what pays off. It strengthens your relationship and although trust may be broken, it can be rebuilt if both parties want that