God the school's response is so sleazy and unapologetic

Some of the big name servos here in Australia have them too, namely Caltex servos.

On the occasion I’ve rushed into an Aldi 30 minutes before closing, they have that too

So you can use KeePass + Syncthing to synchronize the database file across your devices. Keeps it distributed and I've heard a lot of recommendations for this, although I haven't tried it.

If you don't want to do that, Bitwarden is well regarded and probably would suit your needs based on what you've said.

For my threat model, I don't trust any online password manager, so I host my own local Bitwarden server (Vaultwarden) and use Tailscale to securely access it from any device, and if the server goes down, the Bitwarden client keeps a cached copy on the device itself.