Zhenya,

Me too.

I fingerpick almost exclusively now. I also really enjoy jamming with non-dulcimer playing old time musicians. Hearing myself can be a problem in such situations. I could get more volume by strumming, but I don't really do that anymore. Also, non-dulcimer players (at least the ones I play with) tend to get annoyed by the constant whacka whacka of a strummed dulcimer.

I have tried various types of finger picks but do not currently use them for three reasons.
1. Don't care for the clacking either.
2. They change the "geometry" of my fingers substantially. Thus, playing with fingerpicks is always more awkward for me. Playing with fingerpicks is NOT just like playing with fingers, only louder. I figure that if I ever went the fingerpick route, it would have to be an all or nothing proposition. Fingerpicks all the time for at least as long as it takes to become 100% comfortable with them. Not willing to make that commitment yet.
3. If I put them on tight enough to stay on, they are too tight and hurt.

So what do I do?
I have spent years looking for (and finding) loud dulcimers. I have several that hold their own in small groups of guitars, fiddles, and clawhammered banjos. (Nothing can stand up to a bluegrass banjo.)
I take the time to find a "sweet spot" in a jam circle, i.e., the place where I can actually hear myself relatively well and there is a chance that the other musicians can hear me. This obviously depends upon the other instruments and their placement. It is really helpful to play regularly with musicians. They can learn about your sonic limitations and work with you. The most thoughtful ones will actually mute themselves a bit when it's my turn for a break.
I am resigned to the fact that occasionally only me and the musician next to me will be able to hear me.