Apple support tells me that I need to upgrade to Apple Mojave, which doesn't appear to be available. Any useful suggestions? Thank you.

Back up your MacBook Air with Time Machine before you consider upgrading to a newer major version of macOS.

From the menu > About This Mac panel, note the information in parenthesis on your Mac product description line. Then, only using the Safari browser, visit How to get old versions of macOS, and determine if your MacBook Air meets the hardware requirements for Mojave. If so, proceed to click the link to Mojave in the Mac App Store and download it.