new OS not working with my old HP OfficeJetPro 8625. can I switch to an earlier OS?

Okay, let's get real. You've been advised of the tools to do this but it's not a simple matter at all. You will be completely erasing your Mac and starting over. If you don't have a complete and valid backup of your data you could lose everything. You also run the risk of winding up with apps that won't run on Big Sur and require macOS 12 or later.

I would try something before going that drastic. Reset the printing system and then add your printer back.

Reset the printing system on your Mac to solve a problem - Apple Support

Add a printer to your printer list so you can use it on Mac - Apple Support

Update your printer software if your printer has a printing problem on Mac - Apple Support

Scorched Earth Method 

This method will WIPE ALL Data and there is No Recovery - Period.

>> Only works on Intel Based Apple Computer <<

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

The Bootable Installer can Only be performed on an Apple Computer 

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.

Notation: If the computer being used to perform this action is Too New or Too Old to qualify to run the version of macOS - this computer can not be used.

Alternative is, to gain access to a Qualifying Apple Computer from a family member, friend or associate.

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. 

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.

Follow on

Run the Terminal Command below Before attempting anything

 softwareupdate --list-full-installers

You have to Check First if Big Sur 11.7.7 is still available to download

If Yes this terminal command

softwareupdate --fetch-full-installer --full-installer-version 11.7.7

Have you tried removing your existing driver and re-installing from HP, the latest version?

Install HP Easy Start2.13.3.22120910.9 MBFeb 22, 2023 (Download Here)

Taken from: https://support.hp.com/us-en/drivers/selfservice/hp-officejet-pro-8620-e-all-in-one-printer-series/5367611/model/5385111

You can do this: Create a bootable installer for macOS - Apple Support

Create a bootable flash drive for Big Sur. Be certain you backup before starting and read the entire article first.