How to restore old dulcimer?
Instruments- discuss specific features, luthiers, instrument problems & questions
As Paul said it seems your bridge is out!
I agree with all he says but describe things a little differently.
Your scale looks at fist glance to be [Counting in semi tone spacing] .....
2212212221 but afterthe 10th fret it goes awry! [There is no 6 1/2 fret]
Assuming the frets are in the correct relative positions then the bridge position should be set so that the 7th fret is exactly half the vsl [distance between nut and bridge] Then check intonation as string height will cause intonation to go out as you move up the fretboard and increase tension when you depress the string a greater amount. Speaking roughly the harmonic bell note at the 7th fret and the depressed note should sound the same if intonation is correct.
Most dulcimers use a low string height however the gauge of string /tension/ pitch of note determine the physical size of the vibration at the mid point and so string height is a compromise to avoid fret buzz and maintain intonation. If you wish to experiment make a temporary single string bridge slightly lower than the original and find the best intonation /string[bridge height] for your melody string. Old credit cards cd cases video boxes etc provide temporary plasic packers to vary bridge height.
Adjusting at the nut is good but is hard to undo!
Mark your desired bridge position with tape and then you can position the existing bridge once you have altered the height if necessary.
If you are going to reduce the bridge height remove material very gradually from the bottom using sand paper on a flat surface such as an old mirror glass.
Good luck and have patience!