My boys bought me a Gamo Silent Cat for Christmas a couple yeas ago. I have spent more money on pellets than the rifle costs trying to get it to shoot consistent. Same problems that you are experiencing. I can't sell it because it was a gift but pretty much gave up on it. I did buy some heavy H&N Rabbit pellets to try but don't expect any improvement.
A recently purchased $55 Crosman 1322 multi-pump pistol is much more accurate and consistent. Not as powerful as the Gamo but that means little if POI is so erratic.
If you decide to keep it, check into a new trigger from Charlie Da Tuna(sp?). Mine now has a very nice 1.5# trigger but, unfortunately, still inaccurate.