There are several things that could be making it painful to press down the strings. Strings too high off the fretboard ("high action") is the first that comes to mind, and I see you're checking that out. Next, I guess, would be needing a different gauge of string--and you're looking into that, too. I just had my husband lower the action on a dulcimer in preparation for loaning it out to a novice level student, so I was finding this discussion very interesting. Even after it passed the "nickel test" (he didn't know about the dime test), it still seemed too high to me. So I got out a dulcimer that has a very good action (in my opinion--I do a lot of chording) and when I compared the two fretboards, it was obvious right away that my comfortable one had very low fret wires. Was this important or not? That made me think that there could be much more to comfort in pressing down the strings that I had originally thought. A Google search came up with this interesting information on the subject--in the context of guitars, but I think a lot of this applies to a dulcimer fretboard, as well. Here's a link to the "10 Things You Should Know About Frets" article.
http://www.gibson.com/News-Lifestyle/Features/en-us/10-things-you-should-know-about-frets-0705-1.aspx
Additionally, I personally feel the development of callouses helps me play better, even though I know some great players who think that if you have callouses you're doing something wrong. When I was first learning to play, I would put several layers of clear fingernail polish on the part of each finger that pressed on the wire (I would press a chord first, then paint...dry, then press a different chord and repaint, etc.) Those indentations show you exactly where you need the real or fake callouses!
This has helped me; I hope someone else can benefit from my experience. And read this short article--it's very interesting!