The chopper controls are VERY sensitive and pretty different from the controls for planes. If you push forward all the way, it's going to wobble and tumble down eventually. After messing with choppers more during missions, I've found that the controls can actually work very well if you learn to apply as little force as possible on the left stick while flying them. If you want to move forward for example, don't push up, just nudge up a tiny bit and hold it at that position, and hold R2 to to increase altitude as you lose it. If you find yourself dipping too much and want to go higher quickly, push down on the stick. Don't use left and right on the stick to turn, use L1 and R1. That part feels similar to planes at least.
The trick is mostly in understanding how the momentum of the chopper works, and using it to your advantage.
