Need to replace bathroom tub fixtures

cross-posted from: https://slrpnk.net/post/11339091

My bathroom tub fixtures have seen better days. Pretty sure they're original to the house (~60 years). We have fairly hard water, and the prior owners were not the best at maintenance so no clue how long these slow leaks have been going on. But the shower handle is stuck hard on the valve, and the bath spigot is crumbling away and similarly fused in place. Hot and cold come off easily.

Does anyone have any tips for freeing the one handle and spigot? I've tried light tapping with a mallet on the back of the handle to no avail. The spigot seems bound pretty tightly, to the point where I'm concerned about damaging the pipe if I apply too much torque.

As for the leaks, I haven't decided if I'm going to just replace the valves, or try fiddling with/replacing the packing nuts. This is my first time messing with plumbing, but either way seems straightforward enough after watching a few videos.

Any tips/tricks/suggestions appreciated, thanks!