Hello Tom and welcome to the forum.
Problem could be a number of things. I'm not sure what procedure you used to mount the scope, or what scope mounts you are using, but it could very easily be that the scope is slipping. Did you clean the scope tube, rings and the raised scope rail on the rifle really well with some sort of degreasing cleaner? I always clean everything, even the screws/bolts on the mounts, with denatured alcohol in order to remove any and all grease or oil. Then I use a one piece scope mount (usually an Accushot mount) with a stop pin. Then all the bolts get a drop or 2 of blue loc-tite before being mounted and tightened down. I sometimes even run a bead of loc-tite in the dovetail groove where the mount will sit. Then I let everything dry and cure for at least 2 days. Using this process I do not have any problems with scope "creep".
Another thing it might be is simply the pellets. Each rifle definitely has it's own preference when it comes to pellets. The best thing to do is pick yourself up a pellet sampler pack and try a number of different pellets until you find the one that your particular rifle likes. Straighshooters.com has a good sampler pack if I remember correctly. I currently have around 20 different air rilfes, give or take a few, and each one has a preference with respect to what pellet is the best performer for that gun. However, I will say that the pellet I usually start off with, and usually have good results with, is the RWS Superdome. If you can get some give em a try. They'll also do the job on the squirrels.
The last thing it might be is simply your shooting technique. From your post it's easy to see you are extremely familiar with shooting firearms, powder burners that is, but air rifle spring guns require a different technique. Of course I realize everyone has their own techniques for doing everything, including shooting air rifles, but I think you will find that the generally accepted technique is to use a light grip, letting the rifle forearm basically rest in the palm of your hand when you shoot. Try not to rest the rifle forearm directly on a hard surface when shooting at bench or other fixed shooting platform. Try the "easy touch" method and see if it works for you.
The last thing I will mention is the trigger. It's a well know fact that the stock GAMO trigger basically,,, well,,,, SUCKS !! One of the best investments you can make in your 440 would be to replace the trigger with the drop in replacement trigger blade that is available from Bob, aka CharliedaTuna. It's extremely easy to install and I can't even explain how much it will literally transform your gun into a completely different animal. Here's the site.......
http://www.charliedatuna.com/I'm sure I've left out a bunch of good information, but I know some of the other guys on the forum will chime in and add more good info. Don't give up, hang in there. The 440 is a nice gun, accurate and powerful. If you can tame it you'll have a good pest weapon.
Well, gotta run. Welcome to the forum. Look forward to your future posts. Keep us up to date on how you, the 440 and the tree rats are doing. Oh, and by the way,,, we just love to see pics of airgun small game/pest kills. Post em in the "Hunting Gate". Take care.

Jeff