Thank's Stu for the article. I was hesitant to suggest going for the Full Monty to the OP since it's his first time, but eventually we all have to go there.
Marvin, there are numerous times I've completely replaced the packing while in the water
. You cut the rings in advance (using the shaft outside the box as a guide) then open, tear out the old, and replace. Follow Stu's notes if you want to go that far. I haven't tried his drywall screw trick (but will next time I do this), I use a dental pick. The ones the hygienists use to scrape tartar. Nice sharp, but slender hook at the end. Reach in, grab the packing, yank it out.