Martin and Gearoid, thank you for your replies. Apparently I don't have anything setup wrong, it just does not work properly. My host is running RedHat Linux, Apache 1.3.29, and PHP 4.3.4, so there is nothing "Windowish" about any of the software.
My webhost is in the process of implementing something called suPHP that allows the Apache PHP process to run as the same user as the hosting account. So, instead of running as "Nobody", it would be running as "MyAccountName". This is to eliminate the need to 777 data directories (755 will be the new "highest" permission setting) and for the webhost to better identify those who are running "wild" applications.
The reason all of this relates to my problem is that after suPHP, PHP directives in the .htaccess files will not be allowed. Instead, I will be able to include my own php.ini file in my web space with any overides to the main php.ini. Perhaps that will solve the session problem - time will tell.
My webhost will be converting my server to suPHP in a few weeks, so I think I will live with the problem (I don't have any choice anyway) until I try the php.ini solution.
Gearoid, you solution sounds way over my technical head, but if the suPHP thing does not work, then I will be contacting you.
Thanks again to both of you for offereing advice and help.