I stumbled on the solution by accident. And I didn't have to delete any books from the device or from my library.
I prepared to message Sony Support about another issue. To replicate my issue so I could accurately describe it to Sony, I took the following steps. When I completed the steps, my PRS-650 was back to its original fast speeds.
- I connected my PRS-650 to my computer via the USB cable.
- I opened Reader (Sony's Reader software).
- I waited until the PRS-650 battery was fully charged (i.e., until screen said "charging complete").
- I did what I wanted to do in Reader (which was to export an annotation to .rtf)
- I ejected the PRS-650 from within Reader.
- I closed Reader.
- I disconnected the USB cable from the PRS-650.
I noticed that the PRS-650 opened to the main screen a lot more quickly than it had been doing.
The next time I turned on my PRS-650, it took 40 seconds to get to the main screen (not 6 minutes as it used to). I marked one of my books with a highlight, and it took 8 seconds (not 2 minutes as it used to).
During the past 24 hours, I have turned the PRS-650 off and on, and I have made multiple bookmarks and annotations. The speed has remained consistent.
So it seems that having the PRS-650 connected and charging its battery while Reader (Sony's software) is open is what did the trick. I guess the PRS-650 and Reader need to interact every so often. Most of my ebooks have been public domain (so they can be transferred to my device via calibre), which means I rarely had to open Reader during the past little while.