Fyle Typer allows you to change two things as you said, both of them are four letter codes: the application associated with the file ("creator"), and the type of file. This only applies to "classic" MacOS, i.e. up to 9.x
The last four letter code was somewhat of the equivalent of the three letter extension of the DOS/Windows OSs, and was stored in the file's resource fork, this is the reason why extensions were not needed in MacOS up to 9.x. While in MacOS the file itself had the information about what application should open the file this was defined by the registry in Windows, IIRC.
Now with MacOS X (10.x) extensions are used by the system to define what type of file is being used (which for good or bad became the standard for all OSs), MacOS X apps do not create a resource fork and thus do not add the creator and type codes anywhere in the file (which for files created with MacOS X you would notice in older MacOSs by the file not having an icon in the "classic" MacOS finder, and by the system possibly asking you what application to use, unless you have a proper extension and have set up the correspondence between extensions and applications in the File Exchange (formerly PC Exchange) Control Panel).

About Noel's idea: if it was an html file with a link for the MP3 instead of the MP3 itself the file should be only a few KB long, that would be easy to check.

The files "remain true" to WinAmp because the combination of .mp3 extension and the file registry mandate that they are opened by WinAmp (or whatever default MP3 player you have). The information on song title, artist, etc. is not part of the MP3 file per se, it is defined by the ID3 tag, if the tag is not corrupted the information will show, even if the MP3 can not be played. I really have no idea if the ID3 tags can be attached to any type of files that are not MP3s, I suspect it is a possibility, since I have some AIFF (Mac equivalent to WAV) files that I gave a label to and they are always identified by that label in iTunes.
It does sound to me that your MP3 files were corrupted, maybe your best choice is to try a disk repair application in your old computer (if you still have it) to see if the files can be repaired. A possibility would be to just extract the hard drive from the old computer and connect it in your new one and run the application from your new hard drive. I can't think of applications to recommend, what I used to do when I used a Windows 95 computer was to reboot on DOS and run scandisk (you just type that command to launch the program), with the settings for autorepair enabled, this actually saved an extremely crashy computer enough to allow it to boot, install a new hard drive (since the old one had way too many damaged sectors) and recover everything to the new drive by simply copying after booting from the new one.
If you are going to install the old damaged drive in your new computer you may want to back up the new one, just in case the old drive is too corrupted and causes you trouble, and of course the old drive jumper switches have to be set to slave instead of master drive.

Hope this is useful, good luck in recovering your files.
The Time Geek

P.S. IIRC the resource fork in MacOS also allowed to store simple graphic files, this allowed to add custom icons to any file at your leisure by pasting them in the icon window of the File Info window in the Finder (something not as easy in Windows, which IIRC only allowed you to select from a list of icons in a .dll file and not copying-pasting of icons to files). To be fair, I suspect new versions of Windows do allow icon customization. MacOS X still allows custom icons, I have no idea where those custom icons are stored w/o a resource fork, they seem to work but I just lost the interest in applying custom icons willy-nillly as I used to do in college.