I have v3.19 on Ubuntu 22.04 LTS
I saw 3.22 is available and downloaded the file recommended for Ubuntu on the official website (here) and when I attempt to install it shows that it's installed the latest version but when I launch the program it still shows v3.19. I'm not really sure what I'm doing wrong. Do I have to remove the old program entirely and then reinstall to get this to work?

Maybe the setup location has changed?

You can use which tixati to find the full path to tixati binary that starts when you call it from the console (or any other executable). Or just check the full path in process list. You can also use dpkg -S tixati to check for that filename in installed packages. Are you sure that Tixati wasn't running in background when the installation of new version happened?