Reset MacBook Air 2013

as i see it, you have a few options.

1. you could reinstall catalina. if you go that route, the fact that you are installing a clean install of catalina will likely mean that the "troubles" you were having before will be gone.
2. you could create a bootable installer for macOS for your desired macOS. but note that this requires another mac that is capable of running said macOS.
3. depending on which macOS the machine shipped with, you may be able to try entering recovery with the "Shift-Option-Command-R" keyboard shortcut to install the version of macOS that was originally installed on the mac, or the closest one that is still available.

just make sure you are connected to the internet if you try options 1 or 3.

and just FYI, if it was me, i would try option 1 first. :)

The below method will WIPE ALL DATA and there is no Recovery of any Data - period.

To perform this action will require booting from a Bootable Installer. 

This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer. Example : Bootable Installer of Big Sur would have to be done on a computer that Qualifies to run Big Sur.

Once that is done read on for preparing the Destination computer  >>

Only works on Intel Based Apple Computers.

Extra Special Notation regarding the Touch ID equipped Apple Computer

About Startup Security Utility and Must Enable from Recovery Mode the ability to boot from External Drive Before Attempting 

1 - Shutdown computer and disconnect all external drive Except the newly created Bootable Installer.

2- Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive. 

3 - It will present options >> Disk Utilities >> View >> View ALL attached Drives. 

4 - Choose the Upper Most Drive ( not the volumes indented and list below ).

5 - The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as APFS with the GUID Partition Map. This applies to macOS 10.14 Mojave and above. 

5A - Formatting for macOS 10.13 High Sierra and below requires HFS Journaled with the GUID Portion Map.

6 - Once that is done >> backup out of Disk Utilities and choose install macOS. 

7 - Follow the prompts and it may automatically reboot several time. 

8 - Upon a final reboot - Setup Assist will present with the newer version of macOS.

you can follow the steps in this apple support article: What to do before you sell, give away, or trade in your Mac - Apple Support