Do I need to retune when adjusting a spark gap?

["Garry F." <garryfre-at-pacbell-dot-net> (by way of Terry Fritz <twftesla-at-uswest-dot-net>)]

I think I do, although so far, if I don't change anything else but the gap, I
seem to find the best tap position is unchanged.
 
Also, one of the NST's I recently seems to have a dead short in the primary.
Sparks fly out of the wall plug when I plug it in. About half an inch long
sparks, out of the wrong place but it does show promise for a mere 120 volt
wall plug to throw sparks like that. Hmmm, maybe I've stumbled onto a wallplug
to carbon converter circuit?
 
Is there a chance that the primary windings are not a melted blob of metal and
that by rewinding it, I might find the short and fix it?
 
The two I got were out of the same sign unit. The fellow said he had dropped
the unit (Both NST's were inside!) onto a work table. I opened up the dead unit
and the tar inside was had more cracks in it than humpty dumpty!
 
I am just hoping the one I am using isn't humpty's twin except not shorted yet.