I'm pretty sure they aren't samba related, as samba is a protocol, not a file manipulator (it's basically a stand-in for MS's own SMB stuff). The fact that the names are vaguely related to DBIII helps confirm that.
That said, there may be problems with your samba setup, and perhaps those files are a form of DBIII's error correcting/logging system. Or, DBIII might simply be too old to be comfortably compatible with any fairly new networking protocols. On the other hand, you might be facing some sort of virus or system problem, and DBIII is trying to correct for those problems as well.
My suggestion is to look for another solution. I know, it's a hassle, but chances are that you're going to make up for any problems, in the long run. Mysql is probably a fairly natural choice for a migration like this, as you obviously don't need the ultimate DB solution. It's GPL'd, and there are migration tools up the kazoo. It might be a breeze for you to migrate, much easier than someone with something like MS Access, simply because the dbaseII/III/IV stuff is so well understood by so many people.
I know it's a crap answer, but 14 years is 1990, and that means 386-486 days for most systems. Now that Open Source solutions are available, why not take advantage of them? There are other solutions aside from mysql that are just as open, but for simplicity, I think mysql might be the easiest move, and I think the results will surprise you.
Oh, and you probably don't want to try a 486 Linux install, if you're considering that. It's possible, but you have to be a real dedicated Linux user to pull that off with any decent results...