The 'bright' sound is probably not caused by the strings, but by the simple fact that you have a very short VSL instrument which may be tuned to a higher pitch than a standard dulcimer.
What is your current tuning? Try tuning down one notch. If you're tuned GDD or GDg, try FEE or FEf; If you're tuned DAA or DAd, try CGG or Cgc. Be aware that the short VSL Ginger dulcimers often "sing" best when tuned up and bright.
Is your Ginger set up for G or D? If you're set up for G, swap your strings (one at a time) for a the standard McSpad Ginger D strings -- .012 plain steel melody strings, .016 plain steel middle drone, and .026 bass string.
You can certainly buy loose (individual) strings .02" larger than your existing strings without bothering things too much.
updated by @ken-hulme: 12/02/15 10:04:40AM