Thank you so much, everyone--and my apologies for the very late reply. I had a lot of research to do!!!
I bought the Sony Pocket PRS-350 that rcentros linked--thank you!!! I figure it's worth a shot for just $25. I may also try a Kobo Aura, a Nook Glowlight 3 (I'm going to go check them out in store later this week), or a Kobo Aura H2O 2 if the 350 and I don't get along (and if we do, I may also investigate the 650 and T1 models!). I've also decided to try attacking the night reading issue from another angle and bought a $10 "warm" colored reading light. Hopefully that will relieve some eyestrain issues and allow me to keep using unlit readers.
Many, many thanks to rcentros, JSWolf, Hemmi, Barbara1955, kcladyz, Richwood, ZodWallop, and everyone else for their insightful comments!
Also, just to clarify my feelings about Kindle, I was very happy with my Kindle Keyboard models but I had three that all died within a couple of months of the end of their warranties. I don't want to buy used ones and be bracing myself for their eventual deaths! I think the brand is gearing their readers for a very different type of reader than I am, and I'm excited to explore other possibilities.
Thank you again, and now I'm off to research custom firmware and all the other intriguing options folks have mentioned!
May