With the introduction of Catalina, the old /Users/username/Library/StickiesDatabase digest was converted to individual RTFD packages for each sticky item and the StickiesDatabase itself is no longer used. On Monterey, the individual Stickies are found in this folder when you run the Stickies application at least once:
/Users/yourshortname/Library/Containers/com.apple.Stickies/Data/Library/Stickies
If you enter the following in the Finder's Go To Folder dialog (shift+cmd+G) on Monterey:
~/Library/Containers/com.apple.Stickies/Data/Library
It will open a new Finder window on that location, exposing the Stickies folder in it. Copy the Stickies folder to a USB stick for transfer to your new Mac. On that new Mac, once it has been setup with your user account, run the Stickies application once and it will build the above path for you.
Do the Finder's Go To Folder shortcut again and enter the above tilde (~) path, then press return. This will reveal the Stickies Library folder again, with an empty Stickies folder and you replace it with the Stickies folder you placed on the USB Stick.
Run the Stickies application again on the new Mac.