Thanks Robin and Strumelia for the hints. I actually feared something like "recording and critical listening". First I know that I'm not a big singer. Second I can't stand my recorded voice since it sound so unfamiliar.  So I guess I'l stick to practising. Up to now I always tried to sing with the tune. I now started to switch between simultan singing and playing and only playing.  Seems to be good for actually hearing the misstakes.

Concerning the dulcimer: Mine was build by the "Klangwerkstatt" (literaly Sound Workshop)  in Markt Walden, Bavaria. It's a small luthier company (four people working for them including the owner and his wife) who mostly does workshops (from prepared kits as far i know) for building harps, dulcimers, psalterys and other instruments. . Due to Covid19 they can't do it at the moment,  thus they  switched to actually building them. Their webpage has a picture: https://www.klangwerkstatt.de/dulcimer
I changed the tuning to CGG and removed one melody string though. And I  have some extra frets.  

I'm considering to get another one but they also offers diatonic and chromatic epinettes and even a cistera. So: Difficult decision :)
Epinette:
https://www.klangwerkstatt.de/epinette-des-vosges

Cistera: https://www.klangwerkstatt.de/ungarische-zither

Jost, I, also find it difficult to play and sing at the same time.  (I just need to take it slow and keep practicing.)  Keep with playing and singing and having a good time doing so!    

I find it very hard to sing and play at the same time. There are various ways to do it. I've never gotten good at it!
Good for you jumping in Jost.  I like the piece you chose. 
Suggestions?- I think a good way for you to additionally practice is for you to record your dulcimer playing and then practice singing along with it in order to strengthen your singing part. Maybe that sounds counterintuitive when you are trying to play and sing at the same time, but I'm sensing you are having to concentrate so much on your playing here that it may cause the vocal part to wander. Right now your playing is a bit stronger than your singing, so it's hard to run a race with one leg stronger than the other.  ;D

You could maybe slow down the audio of the dulcimer part a little to allow time for you to match the notes and rhythm with your voice. You could use some other very simple tunes to practice matching your voice to each note, even without words- that's a good vocal exercise and after a while it will strengthen your singing!  I remember when I used to practice singing my throat would feel sore... our vocal cords are muscles, after all.

Keep up the good work, and your dulcimer sounds real nice- which dulcimer is that one?