Forgive me for being so stupid here, but I forgot that my Apache server (local to my machine) is not the same as my web host provider's server.
And that makes a world of difference :-)
I finally got Apache to run on my Win Me machine after much tribulation, but run it does now indeed. And I just had a brain fart there as to which "server" I was refering to.
So I guess, viewed in light of your post, that Sessions can replace Cookies without the user having the option to block them? That is what I think could resolve this issue.