Don't know whether I found the problem, but I did manage to get Screen Sharing working reasonably again.
Things had gotten so bad that even mouse moves and keystrokes were getting mangled and delayed, so I decided to try a VNC client (Chicken of the VNC) to see whether things were better. I had to set the host machine to accept VNC connections, so you could say I "touched" the screen sharing panel in System Preferences. I also shut down the Screen Sharing app on the client side so I could use the VNC. The VNC worked OK but would not display the entre screen, so I quit. Then I restarted the Screen Sharing client, and--hey presto--it worked! Response to mouse & keyboard were back, and video was good enough to watch a video.
So, I guess folks could try (1) jiggling something in the Screen Sharing prefs on the host machine, and (2) restarting the Screen Sharing client.
-- rick