So, after my memory worked and told me that the reason was actually the password, and after I searched the net for a new plugin version that wouldn't crash and found nothing, I decided to change the password by myself.
First I looked at the Winamp's plugin folder (Winamp\Plugins) for a configuration file. I was pretty sure the file had to be there, after all, you deploy plugins in Winamp by simply coping the DLL to that folder. And I know some AudioScrobbler files are there - I fumbled with it's cache once back in 2007. However, besides the cache and a log, I couldn't find any files.
Next stop: Windows registry. A quick find for "AudioScrobbler" did the trick, and there it was, a key named "password"... which, my wits told me, ought to contain my password! Fortunately, I remembered the old password: key was hashed and I needed to find which algorithm was used. So I tested a few, producing hashes from my old password: SHA-1, SHA-2, MD5... bang! The key was hashed with MD5. Easy enough.
All I did then was produce a MD5 hash of the new password and replace it in the key. After firing up Winamp, I checked the log to make sure the connection was working again and that's it.
P.S.: I shared this story just to show how we can use our programming knowledge to solve problems without programming and because it was a long time since I last did some dirt trick like this, so I felt a little proud of myself.
P.P.S.: Also, the website linked to above has the source code for a previous version of the plugin. Maybe, if I find the time, I'll update it so it doesn't crashes in Windows 7 x64 and post it in GitHub. I don't know how many people still uses this plugin (or even Winamp for that matter) but I love it and want to share it.