It's a personal thing. I do not like the sound of new steel strings and can't wait for them to be played in. I prefer the tone quality of slightly worn strings without the brashness of newness. I know they're worn out when the harmonics are off or become deader (harder work to play), or when roll-ons need extra effort. Don't know if this would apply to the dulcimer.

New nylon strings are also a pain because they take a few days to settle. I sometimes find with nylons that the wound strings start to unravel slightly and buzz (never happens with wound steel) and so it's time to change. I find nylons keep their tone greatly longer than steel.