Had a similar problem. The thing that was messing mine up was Preferences > User > UI Settings > "Delay Viewer Display by" option. I had it set to 2 frames. Once I turned that off (Unchecked and 0 Frame Offset) the problem went away.
Fairlight seems to have a different interaction with the audio hardware attached to the computer. In my case I'm running an UltraStudio 4k Mini and an RME Fireface 800. Depending on what setting you have under Preferences > System > Video and Audio I/O > Audio I/O > I/O Engine, will determine how the audio hardware will interact with Fairlight. It also seemed to be mitigated if I ran with System Audio, but then the sync is off. So you might experiment with System Audio / Desktop Video as your I/O Engine. But if you're not running an UltraStudio, I don't think you will have this option.
I also highly recommend, if you use an external monitor, to use CatchinSync:
https://quietart.co.nz/catchinsync/This will allow you to see if your video and audio are synced up. Mine were off at least a frame or 2. But the problem with Resolve is that it has a bunch of different delays you can employ and they all affect different pages and parts of the app. So it will be in sync on the Edit page, but out on Fairlight. Super annoying stuff. BMD really needs to rewrite the underlying code, so that all the pages work with the same audio settings.
What BMD do is buyout different software from different vendors and then Frankenstein all of these systems together. When it works it's downright astounding, but at times it's buggy and needs a real architectural underpinning, so that it works more smoothly. But I can't complain too much, Resolve is doing a lot of things right.
Hopefully some of this will help.