Cant access some of a Partition

It appears that one of my hard drives is going to crap. It's my second
(slave) drive, which has 3 partitions on it. 120gb IDE drive. Formatted
to Fat32. This is a Win98se computer.

Oddly, from Dos, I can see all the folders in that partition. From
Windows, I only see about half of them. This partition contains storage
files. Most are backed up, but there is a large folder containing some
valuable info that I do not have backed up. In fact I first noticed this
problem when I wanted to backup that folder to am external HDD.

I ran Scandisk, first the fast method, which told me to use the thorough
method. I ran the thorough one, which took hours. While running, it got
to one cluster and said that one can not be fixed. The rest of that
partition tested ok.

When scandisk ended, it said that it could not fix that one error, and
when I tried to close Scandisk, Windows locked up.

Temporarily, I removed that drive. Since it's all storage, I can still
use the computer. I intend to buy another drive of the same size and
type. However, I really need to save that data, especially that one
large folder, which contains many sub folders.

Like I said, I do not see that folder from Windows, but I can see it
from Dos (I dont understand that). Of course in Dos, all the filenames
are truncated.

I can either stick it back in this computer, or probably connect it via
USB to my XP laptop.

What software can I use to extract the data I want to save? I do have
Partition Magic (not sure if that will help). If there is a bad cluster,
I'd expect a few files getting lost, but I cant see many of the folders
that according to Dos are still there.

Please help!!!

Note: I am not a Linux user, but maybe I could connect this drive to
another computer and boot that computer with some smallish linux such as
Puppy Linux, from a bootable USB. I have used that method to retrieve
data from an XP drive, after the motherboard failed, but I never tried
it to retrieve data from a failing HDD.