I've got it all working now. In case it helps anyone else, I'll post our solution.
As it turns out, it was easy as creating a folder in the root of the site called "sessiontmp" and adding the following line to the "php5.ini" file (may be "php.ini" for others):
session.save_path = /absolute/path/to/root/sessiontmp
Thankfully, everything is working perfectly now. I really appreciate the assistance in getting our problems resolved. As always, the support and community here is wonderful!
All the best!