I have some publicly facing web servers running Debian and Ubuntu. Specifically, what can I do to test and see if my servers are vulnerable?

Debian (trixie kernel) 6.12.113 was allegedly patched for this bug, but there's also a .115 that's been released in the last few days, upgrade to that and reboot.

For Ubuntu (which I don't use), you'd need a 3rd party updated kernel repo, to use a patched kernel from that.

It's not an instantaneous exploit; it apparently takes a lot of iteration over the network. A lot of cloud network providers are mitigating it by dropping options at the border, but unless you test and confirm that, you should assume it's possible for a motivated-enough attacker to compromise your server.