I take that back... I did *not* fix the problem as I indicated in my last post. I just did a fix that happened to work with my testscript. It did not actually end up working on other scripts, however.
Luckily, I did figure out what the ultimate problem was (at least I'm 90 percent sure I did) and so hopefull this will be useful to someone. Maybe I'll put it somewhere in the WIKI. What was happening was that when I went the script ran from cron it would have output that it would want to send somewhere. I had a line in my crontab file...
...that was supposed to "e-mail" me output. Apparently this wasn't working, so it would try to submit output, that output couldn't go anywhere and so the process would quit.
What I did to fix this was something suggested at this website:
>> /dev/null 2>&1
...at the end of each crontab listing. According to the above website, "The >> /dev/null 2>&1 part means to send any standard output to /dev/null (the linux trash can) and to redirect standard error (2) to the same place as the standard output (1). Basically it runs the command without any output to a terminal etc."
So now my scripts are running as they should.