I think I figured it out. I did a fresh install of Debian 11 and installed pi-hole, but still same issue.
The error message mentions the account being locked, so I tried "usermod -U www-data", but that had no effect. So, on a hunch, I added this to /etc/shadow, below the "root" entry (I don't know if order matters, but that's the relative order that www-data exists in my /etc/passwd)
www-data:!:::::::
Tried saving settings, and voila - they worked. I guess the usermod won't add a "locked" entry to /etc/shadow, it'll only lock one that's already there.
Hope that fixes it for you too.