Yes, I do mean that the shifter could be near the end of its life such that it would need to be replaced. The thing about Shimano shifters is that they are pretty much not serviceable (not rebuildable). Don't take it to the shop unless it's a last resort, meaning you've worked on it and you've also replaced the cables and housing. Basically, if the shop works on the shifter and breaks it, you are out of luck, because they aren't really supposed to do any disassembly. It's as risky as doing it yourself. New shifters are expensive (but you may be able to find just the one you need somewhere like eBay).

BUT, you can do a lot of good servicing yourself by flushing out all the gunk. The salty air where you are isn't great for them for sure, but I would definitely try to rehab the shifter before you give up.

It's unlikely you're going to get it off without undoing your tape, because the screw and bracket is covered by tape, and the bracket doesn't really loosen enough (at least on 31.8 bars) to slide off over tape. You can do a lot of work without removing it, though. I had to remove one of mine when the knobby end of the cable got stuck in there this fall after the cable snapped (that was tricky and involved 2 of us going at it with tweezers!) but I was able to do a lot of work with the brifter on the bike.

Just compress the brake lever to expose as much of the innards as possible and flush really well with copious amounts of WD-40 (using high pressure water can also work to get you started, depending on what's up there). I mean DRIPPING with WD-40 sprayed in there at close range. Then move the levers all around and get at any spaces with the spray. From what you said, I'd pay special attention to lubing things up when the inner lever is moved. You might want to take a flashlight and look at the dirt and grime that's up in there when you do this. Sometimes, I also use a Q-tip to get the gunk out of the bigger areas under the shifter.

Wipe down the levers well afterwards with water (they will be slippery with WD40 the next time you ride!), and when everything is dry, relube the shifter with something. You might have to relube a few times over the next several rides.

I doubt you did anything to the shifters when you moved stuff around. It'd be more likely that you damaged the cables then. It's probably just bad timing.

Even though it looked clean then, it could be dry and/or gunky up in there now. My shifters have gotten kind of sticky and unresponsive feeling from time to time. I've only really flushed them once, but I've used a thin cleaning lube (Prolink maybe... maybe not the best, but it's what I had) a few times with good results. Flushing them with WD-40 was a lot better.

It might not be a bad idea to investigate the cable more too. Taking the cables out of the shifter will allow you to move it more freely to clean it. Then you can look to be sure the cables aren't starting to fray or something near the end. This is where you'll run into bartape problems, though.