Air rifles tend to shoot loose so the first thing you want to do is to check all of the screws and make sure they are tight. The two screws in the front of the stock are particularly important, but check them all out. You indicated that you remounted the scope several times, make sure the mounting screws are tight and the scope is not shifting in the mount when you fire it. You might take a pencil and put a line on the scope in the front and real of the rings and see if the scope walks in the rings. Powerful "springers" will destroy the innards of an inexpensive scope, especially one that is not designed for a "springer". Don't know anything about a Gamo scope but I have seen a scope man clamp a rifle securely and then tap the scope and the reticle moved around indicating an internal problem. If you have the equipment you might try that.
Have you changed the type / brand of pellets you are shooting? Try a heavier pellet that will lower your velocity. 1200 FPS will exceed the sound barrier and can screw with stability of the pellet when it breaks the sound barrier and alson comes back below the sound barrier. BTW, PBA pellets which is what Gamo makes the claims of 1200 FPS are not noted for exceptional accuracy from what I have read.
Springers also tend to be sensitive to hold as you indicated. As with a cartridge rifle or handgun, none of the rifle should touch anything hard as it will screw up vibrations and destroy accuracy. Try different holding techniques, pull it in tight, see how it shoots. Hold it looser, see how it shoots, but do each the same each time as you try different holding techniques.
Check the scope for paralax. Clamp it or set it up in a steady rest, put a target out there at 25 yards and then get behind the scope and move your eye left to right without moving / touching the rifle. Do the cross hairs move on the target. If they do you have paralax in the scope and that can impact accuracy if you do not place your head / eye at the same point each time you shoot the rifle. If the scope has an adjustable objective (the front of the scope) try adjusting for zero paralax. If you are good with iron sites you may want to try the rifle without the scope and see how it shoots.
Google air rifles, there are several excellent sites that have a lot of info on air rifles. They can give you some good ideas on what pellets have produced excellent accuracy. I know there is one supplier that sells a combo pack of something like five different pellets with about 25 of each for an individual to try out and see what works best.
There is a guy named Charlie da tuner who specialized in tuning Gamo's. I believe he just retired but still maintains a web site and he might be a source of additional information for you.
Good luck, would like to hear what the problem was if you resolve the issue. I have an RWS springer that is shooting well right now but you never know and it is always noice to hear what worked for someone else.