Initially, I assumed that one possible reason for the poor shifting was that the RSX shifter mechanism was simply wearing out due to its age. At some point recently I discovered that in the last year or two, Shimano has started selling a 3x7 drop bar shift lever set for the first time in many years. The 7 speed Tourney group appears to have been available for a few years and is found on various low-priced bikes, but only last year was a drop bar shifter made available. So, my first solution was to try swapping out the RSX shifters for a brand new set of Shimano Tourney 3x7 shifters.
These shifters sell online for about $90. Amazingly, after buying these I sold off the old RSX shifters on eBay for $65, so this swap ultimately only cost me about $25. I'm constantly amazed how much some vintage bike parts will sell for. The Tourney shifters have the thumb lever like Shimano Sora did until about a year ago. I prefer the "behind the brake" location for the upshift lever, but this works fine. These also have the "indicator" that shows you what gear you're in, another feature I associate with lower end Shimano road groups, and which I always assumed I didn't need. As it turns out, it's actually been nice to have the gear indicator on a tandem because it's not as easy as it is on a single road bike to know how many gears you have left just by looking down.
The Tourney shifters didn't solve my shifting problems. At this point, I started measuring how far the shifter pulls the cable compared to the original shifter that came with this bike, when it finally dawned on me that the rear derailleur on this bike was expecting a different cable pull (known as "actuation ratio") than the shifter was pulling. I had assumed that because the spacing of the 7 rear cogs was identical to Shimano 7 speed systems, a 7 speed Shimano shifter would work with this, without giving any thought to the rear derailleur.
The original derailleur is labeled "Sachs Centera". I can't find any documentation on this old shifting system. Sachs was bought by SRAM in 1997, so I'm assuming this bike and Dual Drive system was made around that time, because I don't think it's much older than that.
Regardless of the technical details, I was pretty sure the derailleur was the problem, but didn't want to spend a lot of money on trial and error. All I needed was a derailleur that's compatible with Shimano road shifters. For $15 on Amazon, I got this "SunRace R80", which claims to be Shimano-compatible.
This is designed for an 8 speed system, but Shimano 7, 8, and 9 speed all use the same cable actuation ratio, just with the clicks spaced closer together, so in theory this derailleur should work with any of them as long as the shifter is compatible with the spacing of the cogs. With the new derailleur installed, I can now finaly reach all 7 gears! Shifting is still a little sluggish sometimes. The derailleur hanger is slightly misaligned, so I'm going to try fixing that, but it could also be that this is just how a $15 derailleur shifts. In any case, I'm happy enough with this setup that I'll stop messing with it for now.