Most planes try to fall out. It wouldn't be much fun if you just put it there and hit the gas. I'd check lateral balance if the fall is 100% of the time to the right, then look at thrust angle changes.

Most planes fall towards the gear and left. This is due to the balance of the plane and torque of the engine.

Just for checking it out, trim the plane with two clicks left rudder and see what happens. That would tell you about warps and the like, but generally will not 100% fix balance of thrust problems.

Hands off hovering looks cool in pictures, but most times a second or two of hands off hovering is all any plane can do. It takes pretty much constant input to hold a hover.