Update:
Sorry, for this very late Update.
Well, the 9-pin Serial to USB Adapter didn't work either in getting the cursor speed, and acceleration to the levels I need.
Thus, my only course was to replace the old trackball.
Amazon had a Black Friday sale on the K64325.
So, I purchased it.
I've got it connected up now, and it works okay.
But I'm going too need to make more adjustments to it.
Plus, the Scroll Ring is kinda funky.
Like it is out of balance, every time I use it.
I'm going to keep my old Trackball as a spare for my older graphics computer.
It is still running XP, and has lot's of older programs that won't run under Win 7 or newer on it.
Signed: Ensign Tzap