It's a 16GB FAT32 formatted thumb drive that should have some 13 GB of my files. However some folders are missing when broswing Windows Explorer. Yet, Explorer says that the device is indeed using 13 GB. How do I recover the files?
First of all you should check if the stick is a genuine one. There are cases where fake sticks are sold that act as if they have the data space available but actually they do not. Search on the web for reports of your manufacturer and model for reports if there were such cases. If yes, your data is most likely gone.
If not, I would create an image with one of the free tools (example) to work on it for recovery first and for safekeeping in case something goes wrong.
Take the image, make a backup of it to keep safe, specially if the image has the size of the files you should have instead of those which are visible. IF it is only as small as the visible files, there is one more indicator that your files might be gone.
Find a tool to mount the image and work first on this one.
I would get a hex editor and see if you can jump to places of the stick and see if you can find indications that the data is actually still there. This depends on what kind of data you have there. If you know file names or even parts of the content such as uncompressed text files will be easier to find than let's say compressed files where you do not even know the name.
Once you have a good indication if your files are actually there, I would get one of the tutorials online to recover lost files such as this one. Work first on the mounted image, if that fails, work on the stick.