I have not seen inherent secure printing in the Apple print panel unless the particular printer driver offers it via the printer specific print panel extended selections (e.g. Job Type -> Normal, Secure Print…, etc.). Xerox does this in the WorkCentre product family. Cannot remember what Konica-Minolta did for this, though they do have a product named Secure Print in the iTunes store to securely print from iOS devices (which may be another issue for you).
After you updated the print drivers for the printers, did you then follow the steps in Reset the Print System so that the new driver information was actually updated for the printer entry in System Preferences : Printers & Scanners panel?
Did the printer vendor driver page actually indicate that the print drivers that you updated were explicitly for High Sierra, and released after High Sierra was introduced in Fall 2017? Installing driver updates prior to this timeframe would implicitly indicate that the drivers are likely not entirely compatible with High Sierra, and that could pose issues.
Finally, for the given Office printer, review their adminstrative guides and see if there are instructions for configuring Secure Print. Is Secure Print a printer vendor extra-cost option that must be installed in the printer before the driver detects it?