Unfortunately, your iMac is "too old" to accept High Sierra. This link shows which Mac models it can be installed on:
https://support.apple.com/kb/SP765?locale=en_US
Sorry for the bad news. The other frustration I feel for you is that unless you have the file "Install macOS Sierra" somewhere, it will be difficult (and possibly impossible) to get it. However, this link looks promising:
https://support.apple.com/en-us/HT208202
Sierra is a very stable OS, and if you want to keep using your iMac for a few more years, that OS should serve you well.
One thing I would recommend is that you do a clean, fresh installation of Sierra, especially if you have (most of the time) just upgraded "in place", ie, allow the new OS to basically overwrite the old one. For that process, however, you will need to have a backup to an external device.