Etydhv

Disclaimer: I know backstories are good and necessary but here's the issue -

Two players have a joint backstory, and one of them is REALLY into it, and it's starting to disrupt game time because he keeps bringing it up. He'll pair off with his "partner" and do stuff just so they can start talking about their shared backstory to impress the other player - like ask him backstory questions he already knows the answer to, just so it gets brought up in front of the party... but there's no way they should expect 4 other players + a DM to listen to this for ten straight minutes (They split off from the party, so I have 4 PCs literally waiting to play).

I want their backstories to be the reason that NEW stuff happens, not the reason new stuff isn't happening.

Last session, the guy starts getting snippy at other players for not listening to him role play drinking a cup of tea while we're packing up cuz the session is over... I don't know how to politely tell him that nobody cares if his eyes turn a different color when he drinks tea. I just don't know how to politely get the game to move on without making him feel like I'm ignoring his backstory entirely, but the truth is nobody cares but it, and he's doing nothing to include the other players in his backstory.

TL;DR - Basically, he wants to write a book and have people read it at play sessions (but he's not a great writer).

As the DM, when one of my players is being boring or distracting, I usually just interrupt and remind everyone about the plot. I'll turn to someone who's not currently engaged and I'll say: "Okay, so you've showed up at the museum, and it's pitch black inside, like someone's using darkness magic. Somewhere inside the building you hear a little boy giggling. There's quite a lot of blood outside the museum, but you don't see any corpses. What do you do?"

(I do this sort of recap frequently, even when nobody's being distracting. I feel that it helps to set the scene and keep the game moving, and also I really enjoy giving these two-sentence dramatic plot summaries.)

Most players take this pretty well -- they understand that it's the DM's job to keep the game moving. Ideally, when I remind my group of the problem they're supposed to be solving, it even gets the annoying player to focus on the problem instead of whatever side path he's focusing on.

It's also worth noting: it sounds like you have six players, which is really too many -- most groups are best with four, for pretty much the reasons you're describing. (When there are six players, each player only gets to talk one-sixth of the time, which is not enough.) It's probably not necessary to ask your problem player to leave, but if you did have to drop to five players instead of six, it would leave you with a better game.

This is a great opportunity to use the incentive of INSPIRATION POINTS. Tell everyone that you are awarding inspiration points to any and all that help keep the story moving. In this way you are not the only one telling them to stop but are encouraging others to talk over them and play forward through the distraction.

I was GMing for a group of 4. One of the players ran a wizard that would continually get in trouble during the course of the fight because she wasn't learning her spells. Every fight was either Bigby's Hand with fireballs or Wall of fire with fire balls and the group used no tactics. The cleric wasn't using her healing word as a bonus action spell. At the end of one of the sessions I encouraged everyone to get to know their characters and use their imaginations to solve their problems. The first time I saw the wizard cast fly in the beginning of the round to avoid being lunch meet for lindworms I gave her an inspiration point. Similarly, when the cleric decided to use healing word as a bonus action I gave her an inspiration point. After a few sessions I could pull back from awarding inspiration points because this way of playing became the norm. They learned the way of playing that I was trying to direct them towards and they adopted it as their own.

Similarly, as you encourage your players to assist in moving the game forward this will become their new way of playing as this behavior is reinforced and the other is discouraged.

I must admit that I have not had to use inspiration points for behavior problems. Although I am a teacher and use reinforcements for my students to learn replacement behaviors. There is plenty of peer review case studies on that