Upgrade OS or not?

How old??

If your iMac is capable of Mojave, then you have a model newer than the Late 2011 model. That being said, anything between then and the Late 2013 model can be upgraded but only as far as one more OS to Catalina, which itself is getting long in the tooth. The Mid-2014 to Mid-2015 can go to Big Sur, and the late 2015's to Monterey.

Rest assured, your old files will be fine, but still, take the advice to make sure you have backups before you do anything. What you could do (if you intend to stay with this particular iMac) is buy an external SSD and install the newer OS on that and Migrate your current system over to it. Two benefits: 1) being an SSD, it wll run circles around your existing iMac's harddrive, and 2) you willl have your internal drive intact at Mojave in case something goes south. Alternatively, you can get Apple to swap out your internal for a new SSD (I wouldn't do this yourself as the iMac's are particularly fragile to crack open), and put your old drive into an external enclosure as your Mojave backup, which you can always boot to if some of your other software can't/doesn't run in newer OS.

If you decide to update your OS,  make at least a Time Machine backup just before you update, so you can revert to you current system in case you regret the update.

I wouldn't rely on customer service for advice. They are committed to getting you running the latest software, without much regard for the effect on your system, or what you prefer to run.  Why are you considering any upgrade at all? For sure, one day your Mac will die and you will need to, but if it were me I'd put off the evil day. In the worst case you won't be able to update to Big Sur, you get stuck on Catalina, your old apps won't work, the new apps won't run, and Adobe won't give you the apps in between. Dead in the water.