Hi Susan -- yes -- post pictures, please.  A good close up of the whole tuning head and nut would be just right...  That way we can be certain to get our answer(s) right

What you're referring to as "fine tuners" may simply be ordinary geared tuners with much higher gear ratios -- it takes many turns of the knob to make one full turn of the shaft.  That's  a GOOD thing.

In general  -- if you have 4 strings: With the tuning head to you left (assuming you're right handed) the bass string should attach to the far side tuner shaft closest to the fretboard.  The middle drone string attaches to the other far side tuner shaft, farthest from the fretboard.  On the side closest to you, the  nearest melody string attaches to the nearest tuner shaft and the other melody string attaches to the remaining shaft.

The attached picture from Folkcraft shows where 4 strings should lead.  Whether a flathead as shown, or a scroll head,the relative locations are the same...

 

 

Hi, Susan, your instrument sounds to be wonderfully made!  Is there some way you can post photos here so, perhaps, better advice can be offered? 

Hello, I have purchased a 1978 Anne and Ed Damm Mountain Dulcimer #198.  The tuners are unlike any I've had before, and I'm wondering if there are some tips and tricks to using them properly and restringing.  They seem to be what I would call a sort of "fine-tuner."  When winding the string on, there is very little movement.  I'm not even certain what order the strings should attach to the pegs.  Thank you for any assistance you might be able to offer.  
This is a really fine dulcimer and appears to be made from pecan.  It has lovely tone, no damage, and I'm excited to start playing!

updated by @susan-mackenzie: 01/17/20 11:58:08PM