Well, I cleaned the track pad several times, ran Disk Utility several times (once from the OS operating disk), and did everything I could think of or that has been suggested.
Nothing fixed the problem, although it was intermittent - sometimes hardly a problem but other times so bad that I couldn't type a short sentence without typing each letter slowly, watching to make sure that the letters were moving forward (not skipping around all over the place) and that nothing had been highlighted (which would mean the next key stroke would delete everything).

Finally, I took it into my local Mac shop and it turns out the culprit was my battery, which had been overheating. (I had noticed the battery cover bulging in the last 2 or 3 months instead of being flat against the rest of the laptop case.)   Got a new battery and now everything works just fine.