You can't change other people. You can only change what you do.

It sounds like you should just block your sister on social media and ignore her otherwise. She's making this very much about her and every argument verifies that she has a legit claim for her input. Once you stop talking to her about this and really don't know what she's talking about, she'll look like a raving mad woman talking to herself. Talking about a couple that sleeps together.... what's new or scandalous about that? It only makes her look dumb in front of others.

You're so busy fighting your sister, you haven't had the time to find out if you really like your gf. The first months are tricky, so your sister is actually doing herself a disservice. Perhaps you should point that out to her before stopping communication about this topic: every time you try to cause a fight, you unite us more.

Perhaps it's time to start looking for ways to move out and get a space of your own. Jobs and roommates and such. No, don't move in with the gf for a while. It's a bit of a long term plan thing, but it's something to start planning now.