It looks like you have one of Delta's older 3530 models. Looking at the exploded diagram for this model shows part # H24 (the handle covers) with only a set screw holding it in place. If you've already removed that and the cover still won't lift off, it may be bound up on the stem from having been turned too far in one direction or the other (if it was leaking, I could imagine someone may have tried extra-hard to turn it off). I would try slowly but forcefully turning the faucet handle past its resistance point, with it in the on position first, then if that doesn't loosen it, try doing the same but the other direction with it in the off position. You may have to alternate this a few times to get it to free itself from the teeth on the stem.
If none of that works, there is a retaining nut on the underside of the sink counter (part # RP9519 in the diagram). It's not a very thick nut, so you'll have to put your hand right against the underside of the counter top to feel it. The threaded part of the stem assembly likely extends well past the nut, you'll need to make many turns to get it off. Taking the handle out from the counter will let you examine it in more detail and you can take tools to the less attractive parts to gain torque, hopefully avoiding any more scratches on the handle itself. Just be careful not to grip too tightly on the screw threads or you won't be able to reattach it to the counter!
According to your picture that valve looks like it has flow controls on each line cold and hot - turn the screw on the cold line - CCW a little bit and see if you now have cold water (be careful as the screw could come all the way out and you will have full on water ).
It could also be that when the shower lines were installed that the installer sweated the copper joints and it got clogged with some debris (not unusual). You would need to do two things here :
1: Remove your flow control on the cold water line - that screw and nut assembly next to the cartridge (with the main water valve turned off of course).
Verify you have water there and check for debris which could easily come from someones copper pipe solder sweating job.
Have someone turn the main on ever so slightly - verify you have cold running water coming through to the shower valve.. it might also force out some debris.
If you have Water flowing - re-install the flow control valve - see if you have cold water from the shower valve ..
2: If no cold water is coming out of the shower at this point and you have cold water at the flow control - you will need to remove the cartridge and check for debris there and also that the cartridge is properly installed.
Just one other comment here - the pipe fitting for the shower cartridge is designed to be screwed into a supporting board - I would try to install a 1 by 3 or 1 by 4 in the back of that flange between it and the drywall and lengthwise between the studs - so my pipe would be supported and locked down - no movement and flexing..
Best Answer
There are manufactured solutions for your problem that will work way better than glue/epoxy or a bunch of work with saws and stuff.
Use a "fit-all" handle arbor that attaches to the stripped stem with set-screws. There are dozens of handle styles to choose from.
Will it prevent you from having to find a replacement stem or replace the whole faucet? YES
Will it work way better than other options and still be aesthetically pleasing? YES
Is it simple and easy and quick and relatively inexpensive? YES
Will you need to change both of your existing handles/knobs? YES