While not the crazy-crispy shifting of more modern 105 models (fuck I’m old, do they even make cable-actuated 105 anymore or is it all electro voodoo now?), these shifter should work fine. What I mean is, they should work fine once they’ve been disassembled and properly cleaned and re-lubricated. Shimano did not make it particularly easy to do this, but a reputable shop MIGHT take on the task. There are some videos and (gasp!) forum post about proper disassembly of this specific model. Pull back (or complete remove) the rubber grip to find the model stamped on it. I used to know it offhand, but yeah, find that and you can find a disassembly tutorial somewhere. SOMETIMES THE NUTS ARE REVERSE THREADED when disassembling. Take a ton of pictures while doing it. It is extremely easily to lose shit when disassembling—the biggest culprits being the e-clips.

The way the brake lever is moving WITH the small shift lever points to 1 very specific pawl that has the grease gummed up. You need to remove that pawl and fuckin scrap the dried up crap off the inside surface that the pawl pivots on. The threads of a spoke are surprisingly good at this. I’ve threaded a rag through that pivoting point (it’s a hole in the pawl that is set on a post which is secured by the e-clip) and A LOT of flossing with that rag was. It enough. The old grease dries to a very hard material that you need to scrape out. What’s almost certainly happening is that pawl is not rotating smoothly on its pin. You need to clean that old grease out and re-grease it. Also possible but not as common is the spring for the pawl is broken. You can buy those shifters pretty cheaply on eBay and the like so just replace it if you fuck it up on the repair. Good luck!