Sounds like you’ve got some serious stick drifting issues. There’s a couple solutions that don’t involve soldering. I’ve had this problem happen with me multiple times and just recently found the first solution usually works for me.
1st (and easiest). Try getting a spray can of “Electrical Contact Cleaner” and just spraying just a little of it into the base of the problematic stick, pushing the stick away from the nozzle head and rotating the stick multiple times to work it in. I think that this mainly fixed the problem for me because the Switch Pro Controller has a major dust/grinding issue with the sticks and this spray is a good and safe way to clean all that dust. It should only take a couple of seconds to dry.
2nd is to open up the controller and replace the sticks entirely. This isn’t nearly as difficult as it sounds. Fortunately, the Pro Controller uses regular Philips head screws, so you just undo all the screws (careful not to lose them!) until you get the front plate off (carefully!), then just pop the sticks off and put some new ones one! You should be able to find some replacement sticks with Google pretty easily, but these metal ones here https://smile.amazon.com/dp/B01M32MP9G/ that include replaceable heads and a tool kit to get into the controller don’t grind dust up like the base model and work just as good. Maybe while you’re still in the controller though, try spraying some electrical contact cleaner (from a safer distance this time) around the board to get some of that dust that has surely accumulated from the sticks.
If that doesn’t fix it, you likely have a malfunctioning joystick component on the controller’s circuit board, and repairing that (while not terribly expensive) requires soldering, which I have no experience with. I know you posted this a long time ago, but I hope this still helps.
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