So there's a lot of decent suggestions. Unfortunately, the net cost for basically everything is going to be ~$200. For example, a decent bike fitting at a different shop would probably cost $100-150 + $20 for a different stem. That said, getting a a proper fitting is probably the best solution. Almost all of your issues are likely caused by a poor fit.

Interrupter levers aren't really a solution, they're more of a work-around to a poor fit.

So a proper fitting is almost certainly the best answer. A secondary answer would be to start meeting with a local club, just to see if people can spot anything obvious. You could maybe combine that with a $30 adjustable stem to self-fit. Keep in mind, self-fitting is going to require several rides and best case gets you a position that a proper fitting would have found in an hour or less.

Your hand/wrist pain is usually caused by having too much weight on your hands and/or a death-grip on the handle bars. How you position your hands on the bars makes a big difference, you want weight on the meaty parts of your hand not the middle of your palms. Unfortunately, the ergonomics of Sora levers are very good for braking from the hoods, especially if you have smaller hands.