The HP Laserjet 3600N support page at HP only lists Mac OS X 10.6 as being supported for this printer.
And the last set of firmware and drivers were released in 2006, so you are very lucky it has been printing
all these years. Click on the link below.
https://support.hp.com/us-en/drivers/selfservice/hp-color-laserjet-3600-printer-series/501044/model/501046
You could try downloading and installing the mac firmware that is offered.
Once installed reset your printing system.
Reset the Mac printing system to solve a problem - Apple Support
Also try the HP Smart app available through the App Store download and run
it should walk you through setting your printer up and check for any new software,
but with such an old printer it may not.
If that is the case you will need to ask HP.