Fiddle
Adventures with 'other' instruments...
The general advice is NEVER TAKE ALL THE STRINGS OFF A FIDDLE AT THE SAME TIME. (Unless you are a luthier doing a major repair.)
String tension holds the sound post in place. You place the sound post and, carefully position it, with all strings loose. Tightening strings without a soundpost can break the face.
Before they were installed, the pegs were probably all the same. During set-up or a service, individual holes may have have been reamed and pegs smoothed with a sort-of pencil sharpener to improve fit and smooth turning.
I'd suggest looking at http://www.makingtheviolin.com/ before fiddling with fiddles. There are other useful violin sites, of course.
Actually "one string at a time" is good general advice when replacing strings on any string instrument, including dulcimers.

There's hope for me yet!!