Well, the very kind Michael Hettrick has sent my order of Teflon gaskets for the beast and tomorrow I'll go about replacing all the Viton gaskets. The main seal will still be copper, and there will be the one Viton gasket at the base. I'm hoping that since this gasket is at the base, next to the pump, it won't have a huge effect on the pressure of the system - we'll see. Currently I can get the base pressure down to 10-6 Torr. If I can get an order of magnitude decrease in the base pressure, then I should be doing okay.
Moving forward, at least. :)

Keep us up to date on your latest, and thanks for all of the useful info.
How do I get ahold of Mike Hettrick?
Any help would be appreciated.
Thanks.
Dick Merk