I am very interested in subbing to the p2p server.
Good choice.
I've been asking around on the f2p server and on some livestreams, and everyone seems to say that the p2p server is completely empty.
Completely is not the correct term. Incredibly would be a better term. In relation to the F2P server, P2P would feel "dead." There are very few people in the leveling zones, and there are only a few handfuls of people at end-game. But, those that can stick it out and can get "in with the in crowd" seem to have a lot of fun and swear by the server as heaven's gift to this world.
Is the p2p server worth investing?
To some, yes, to some no. It's completely your choice and your play style. If you're able to grind, you can play either server, as both servers require time. If you have less time but have a bit of extra cash you can throw towards the game every now and again, I would suggest sticking to the F2P server. If you have very limited cash and lots of time, get your ass on the P2P server, because you will be on a MUCH more even playing field than on the F2P server.
I really want to start fresh on the p2p server.
Starting fresh is always a scary thing. It's a mystery as to whether your investment is going to be worth it and unfortunately, you'll never know unless you try. I would highly suggest giving it a shot.
Not having fun with my situation on f2p and my character.
I'm sorry about that. To be fair, even I, who has almost every CS item in the game doesn't find it all that exciting right now. The game is in a pretty big lull, this patch is probably one of the worst they've ever come out with. Just keep hanging on and some day we'll have 7.0 which will be a huge change to the game, hopefully for the best.
Good luck. I hope this helped at least somewhat.