My iMac isn't letting me use target display mode with my new MacBook Air.

In short, please read Target Display Mode, specifically where it says:

iMac models introduced in 2011, 2012, 2013, and mid 2014

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.
• The cable connecting the two Mac computers must be a Thunderbolt  or Thunderbolt 2  cable.

This simply means your new MBA is too new. I strongly recommend that if you want/need an external display for your MBA to buy one, this is much easier to setup, much more reliable and does not require expensive cabling.

If you read the article linked above your name, you will see the VIDEO SOURCE Mac— your Macbook Air—must be 2019 or older AND running no higher macOS than 10.15 Catalina.

You will also see that the iMac must be Late 2009-2010 (27-inch) or Mid 2011 - Mid 2014 (all sizes) and running High Sierra or lower, to act as a monitor for another Mac.

TDM is dead technology.