The nitro gun is still a "springer", and if anything it's actually harder on scopes than a coil spring gun. Point being all those negative issues apply, and some.
If you had to adjust the elevation counter clockwise to compensate for the barrel pointing down, then you're reducing the tension of the spring on the internal erector tube. The reverse recoil of the gun, which isn't all that strong but it's very high G, can move the erector tube around from shot to shot. Picture hitting the gun from the rear with a hammer each time it's fired, it shakes things up and can even break weaker parts.
When the erector tube jumps from that shock it normally settles back down in the same spot, but not always, and the weaker the spring pressure on the tube the less likely it will. Other factors no doubt contribute, like the roughness of the contact surfaces between the W/E screws and the tube, the friction at the pivot point, and lube which I doubt it has any. If for example friction and stiction were minimal and everything greased it would likely settle correctly each time, or be far more likely to, but do you want that or do you want it to stay put in the first place? This high G shock is also what breaks scopes, makes them move on the scope rail and/or scope in the mount. It is also what breaks picatinny rails off guns. The scope that came with the gun is low quality as you can tell, so while they can work for most things and are kinda a quality match to the gun, don't expect much from it.
So, I'd shim the scope so it has adequate tension on the inner tube. Basically adj the W/E screws to center so the inner tube is more or less centered, then shim until close to on target, then adj the W/E screws to dial it in. This centering of the scopes inner tube also aligns the lenses which helps accuracy a bit.
You can also make the gun less abusive to the scope in the first place by reducing this reverse recoil shock. You can do this by fixing any leaks it has, and increasing the compression ratio of the piston. It's kinda like a car engine and you have leaky rings and a dished piston. You can buy a better/stronger scope, but imo you should fix the issues with the gun as best you can before subjecting a new scope to it. The fixes also bump power and accuracy so why not? In 22 I imagine it's a bit low in the fps dept so all the more reason to do this.
Pellets can be a factor in scope abuse too, mainly the the weight, but the type and brand can affect accuracy quite a bit so best to get ones that are both great quality and a match for the guns power. If you, or anyone, wants info on how to improve this and similar guns you can write me at chevota at hotmail for a free guide with pix and arrows etc. It's over 7meg and includes ~100 pix which is why I can't post it here.
The guns also have other issues which cause inaccuracy, like the barrel lockup which if you'll check allows the barrel to move side to side and often even up/down. The rifling/bore quality is what anyone into firearms would consider completely unsat. This and more info is in the guide. Just let me know what gun you have, and "Nitro Trail" is not enough as there are at least four different ones plus various stock options. The guide also includes some trigger fixes, not the common washer mod but several options for various skill and ambition levels. If people insist on the washer mod I can explain how to make it better if needed.