All I can suggest is what worked for me on my macbook air. After trying many approaches, I eventually reverted my macbook to the previous OS level (Mojave?) using a downloaded image on a flash drive. I don't recall if I first reverted to factory settings. However. I ignored the screen and just pressed the appropriate keystrokes to do a boot from a drive. then I restored from time machine (IF - the old timemachine backup wasn't overwritten after your Catalina install...).
Anyway, once I had the computer in a roughly pre-Catalina state, with the screen working, I deleted/removed ALL 32 bit applications (including uninstall packages for Adobe), and in fact took off all apps that might conflict with Catalina. I think I removed all extensions too. I then took a time machine backup. Then I downloaded Catalina and did a clean install. After it appears stable, reinstall any non-32-bit apps you might need. That worked. And I haven't had any issues with Catalina on my MacBook Air since then.
If you check through this whole discussion, you will find the exact steps I took. BTW - my Macbook Air is late 2012 I think - anyway, it was listed as able to upgrade to catalina. I think I removed any partitioning that Catalina might have done as part of this process to reinstall Mojave.
Hope that helps.
DON'T pay for a new screen - the Apple support folks don't know what they are talking about, and never joined in this discussion to my knowledge.