We just installed a new shared printer. When we select this printer in SalonTouch from the dropdown list and hit save, it does not save this setting.

The printer name that you see on the workstation is probably too long. Check the character length of the network path plus printer name (example \\server\star scp 700 (receipt station) line mode printer ). If it exceeds 50 characters then SalonTouch is not able to save this setting in the Database.  Rename the printer to a shorter name less than 50 characters.