It's really hard to comment on one side of a story in a relationship you're not familiar with, but based on what you're saying, I'd say your girlfriend needs to grow up and evaluate her insecurity and trust issues.
Being in a long-distance relationship is hard enough. It's absolutely miserable from the start not being able to be together, and if you're not in it together, if one or both of you are working at odds with the other, it's nigh on impossible.
I know from experience it's hard to trust from (in my case) 5,000 miles away. But frankly, if she can't trust you now - and if she keeps demonstrating you can't trust her (like to not violate your accounts) - neither of you are going to be able to trust each other even when/if you live in the same house.
There's no easy way around the trust issue. It sounds like your girlfriend has a load of insecurities, and that's not something that's going to go away over night. But she needs to at least acknowledge that she's being ridiculously unfair by behaving the way she is with your accounts, and not being willing to share her own. Even if, at some point, you've given her reason to doubt your fidelity, she has to decide whether she can trust you without being able to control your online life, because try as she might, she can't remove girls from the world. And social media isn't the only way to meet them.
Like I said earlier, I don't know your relationship, and I don't know her side of the story, so it's hard really to tell you what to do, but if I were in your shoes, I would put my foot down about the password thing. Generally, it's a terrible idea to share your password with anyone, but especially with someone who has demonstrated repeatedly they will abuse it. Change it again, and don't let her have access to it. She has violated your trust just as much as she seems to think you're going to violate hers.
Otherwise, you're in a relationship with extremely unhealthy boundaries, and I guarantee you, closing the distance will not change that. The nature of the boundaries may change, but the unhealthiness will persist until you both agree the behaviour is unacceptable.