I just clicked on Update in the Software Update window, which just said HP Printer Software Update, no specific drivers. The HP Designjet T610 24 inch printer package version which appears in the Printers folder in the Library folder is 7.1, with a creation date of 8/4/11. Interestingly enough, Time Machine shows the same version and date going all the way back to the earliest instance in November 2011.
I was getting all ready to do some multi-step Terminal thing listed elsewhere in the forum for restoring a printer driver from Time Machine, but now I don't know if that is even the issue.
I fixed it by doing the following:
Deleted the printers in the System Preferences: Print & Scan window.
Deleted the hp folder in my hard drive root directory: Library: Printers folder
Deleted all the HP PPDs in hard drive root directory: Library: Printers: PPDs: Contents: Resources folder
I then downloaded the latest HP printer software update from the Apple site, ran its installer, and installed my HP printers again in System Preference: Print & Scan.
Success! I will need to recreate some custom page sizes in the DesignJet driver, but at least printing works properly.
By allowing the Apple Software Update to install itself from the App Store Software Update window, I must have allowed some sort of corruption of the printer software to occur.