I believe the only thing that's pushing the rear end up is the spring, and everything else is to do with dampening. I.e. the shock will slow down the rate of compression and rebound, but it in itself is not going to exert any force in a neutral state
So it could be a worn out spring (unlikely) or a flogged bearing
Quite simple to diagnose, get the bike off the ground. Grab the wheel and shake it left to right from both the sides and the top/bottom. Do the same with the swingarm itself, as well as the shock and linkage. Remove pieces and repeat. You'll soon find if a bearing is cactus and causing extra movement, you'll also know if they are stiff/notchy once you remove the wheel and the shock and take everything through its range of motion without resistance. Everything should be SMOOTH. Swingarm should not hold itself up for instance, it should smoothly move from bottom to top, and drop to the bottom when you let it go, same with all the linkage parts. Of course if they are smooth but flop around then that too is bad
Either way, it doesn't take long to remove a swingarm and give everything a good greasing. Even removing bearings isn't too bad as long as the bikes not too old. It sucks when they are rusted in.....
Whatever the case is, I love regreasing bearings. They make everything so much smoother and generally nice. You can tell a HUGE difference between a bike that see's a bit of grease and basic care. Friends Dpro 140 only has 16 hours on it and the rear end felt CRAP. Very harsh and resisted compression and rebound. Removed and regreased everything on the rear, now its fantastic and feels so much better bouncing up and down. There's a distinct lack of grease on any bike, but chinese ones are... yeah... what grease?
It's always best to do this when the bikes brand new before everything starts to really set. Axle's for instance, some people say don't grease them, I say they're silly. A greased axle slides in and out easily with just hand strength. After a few months/years if they havn't seen grease they're a right prick to remove and often need a big hammer. Same thing with bearings. It's MUCH easier to remove bearings that have had the outer races greased, than rusted in ones