Normally, Bitwarden automatically logs out (deauthorizes) all your sessions when you do a master password change (usually within an hour of the change), so it is critically important that you disconnect your devices that still have logged-in Bitwarden clients from the internet — immediately!
As long as the device is off-line, it will not receive notice about the session de-authorization, and you should be able to remain logged-in long enough to do a vault export as explained below).
You should be able to by-pass the master password requirement for vault exports, if you feel comfortable using DevTools. I have provided a set of instructions here — the instructions are from 2024, so a few details will have changed since then, but users have reported as recently as last month that the procedure still works, with minor adjustments to some of the line numbers (see here and here).
There is also a simpler technique, linked in the P.S. of my detailed post. However, if you are using a non-Chrome browser (specifically, browsers still using “Manifest v2”), then you will have to refer to the follow-up discussion just after my original post, and read the instructions provided here.
If you let me know what browser and operating system you are using, I might be able to provide more specific instructions for your extension.
Is it possible that you did not select the correct server region (bitwarden.com vs. bitwarden.eu) while on the login form?
Other than that, I would strongly recommend that you contact support to check if there could have been a back-end corruption of your account data. If they find such an issue, then they should be able to restore your account from a backup (but only if it has been less than 7 days since the password change).