Even if you didn't you still need to retrieve the user and password from some storage to validate the key, which invalidates the reason for JWTs in the first place, since you supposed to be able to validate them without access to an auth service/db
And now you're starting to understand why JWT's aren't worthy of the hype. A truly stateless JWT implementation is insecure since you can't invalidate existing tokens.
The usual compromise is make JWTs have a short (<= 15 min) expiration and provide the client with a refresh token that doesn't expire, but is stored server-side. When the user logs out, the refresh token is invalidated server-side so a new JWT can't be issued. You're re-introducing state with this solution, but you're making it so you don't have to bang on a database with EVERY request, just one every ~15 min or whatever your expiration window is.
I.e. if the user's password has been compromised, then the attacker with the password could have started a new session using it. The question is, how do you (allow the user to) revoke the auth that was granted by the compromised password and invalidate that fraudulent session.