If playing in a system that explicitly rewards teamwork hasn't done the trick, then your group just might not have the buy-in for that kind of game. It sounds like playing for the drama is what everyone wants. Why not channel this tendency into a structure that rewards such play with an upward spiral of fun?
In-fighting can certainly be played in a way that isn't self-destructive. Partly you have to let go of the assumption that there is a party that needs cohering. With the right GM tools, you can run a four-way split group of characters without anyone getting bored. Running a game with a number of independently-acting PCs can be very rewarding, and is excellent for putting the focus on (fun) constantly-shifting conflicts and alliances.
There are two games that I'm familiar with that reward PVP action with not only the immediate pleasure of vying against another player, but also with constructive consequences and complications: Burning Wheel and Apocalypse World.
Now, I'm not suggesting that you should be switching systems. You might already have a game that you really want to play. Besides, you're asking for GM techniques rather than a system recommendation. However, it sounds like your group has a willingness to try new games, so my suggestion is to play one or two sessions of one of these games as a meta–GM-technique:
Play one or two sessions of a game that 1) supports long-form play, and 2) either encourages or is perfectly happy to accommodate PVP play. The point is to experience a system that both allows for internecine conflict in your group and is designed to not collapse under the weight of the PCs' differences. Apocalypse World would be my first choice because it's more a GM manual on how to run exactly that kind of game, and it just happens to also be a playable game so it can demonstrate the techniques.
See how the game ticks. Study how the in-game situations emerge and are channeled into productive, rather than self-destructive, play. See if you can identify the exact GM techniques that are encoded in the rules that make this happen.
Play a game of your choice, incorporating what you've learned from the other game. This will probably not work the first time, but the point of the exercise now is to see what you have to do as a GM to get the productive play you want out of PVP &etc without the game's rules doing the heavy lifting. If you can re-implement the technique in the absence of the rules, then you've learned a technique that is portable.
These two systems are also, incidentally, perfectly suited to non-PVP play, so your group can find its sweet spot organically within a single system. Hopefully that will give you the most "natural" view of your group's dynamic, when they're not fighting the system to get the drama they crave, and where the system/player friction isn't pulling the game down into an un-fun spiral. Of course these aren't the only games that will do it, but they're the ones I know that would work for such a study. And hey, there's also the chance that they'll do just what you want and your group will want to stick with it.