Gary,
I'm glad you've found a sport you can practice despite your disability. I've jury rigged a shooting lane in my basement, and we've got a few members who shoot in the hallways of their apartments. Indoor shooting is a great advantage of the spring rifle!
Springers are what is know as "hold sensitive". That means that any difference in the way you hold the rifle from shot to shot will result in a change of POI from shot to shot. Your old pumper wasn't a springer, and wasn't as hold sensitive.
The hold that works best for most spring rifles is to place your non-shooting hand under the stock about where the rifle will balance. That's usually a couple inches ahead of the trigger guard, and may feel quite close to a firearm user, but you'll get used to it. That hand also does not grip the stock, it's open flat, and the stock just lays in it. The shooting hand grips the pistol grip only enough to guide the gun, and the gun is lightly shouldered (touching, not buried!). I read on Tom Gaylord's blog, and found it to be true, that the best way to shoot good groups is to get into your hold, get all ready, aim at the bullseye... Now, close your eyes and count to 5... Still on target? If so, shoot. If you drifted, you're forcing yourself on target, so try to reset so that just slumped there, you're on. The goal is for the shooter to be almost dead weight behind the gun, a mass that recoils the same each time.
Then it's just up to you to pull that trigger ever so carefully.... Sweetening that trigger up will help some, but with practice, many find those Gamo triggers wear in to be smoother, but they're allways heavy. Charlie's trigger gets glowing reviews, and anyone can install it!
The hold sensitivity thing makes it tricky to bench a springer. Most find the best results come if the gun is still only touched by the shooter (not on a bench pillow), and the shooter's hand rests on the pillow. I think this is because the pillow responds differently to the recoil than the pillow does.
Follow thru, as you know, is absolutely key. The recoil is done so long (in gun terms) before the pellet leaves the barrel, it's important to stay on target till it's all over.
What a great sport we have in airgunning! One could start shooting springers as a child, and still have stuff to learn and room to improve when he's enjoying it in his retirement!
Have fun, Gary, and be sure your backstop is up to the task! Even 4" of lumber will be blown thru if you're target shooting long enough....
J