There is also a shortage of organists here in parts of the USA. Some places seem to have too many organists, like Boston, but where I work, in New Jersey, there are not enough. One good effect of this is that at least in the larger churches, salaries have gone up because they have a hard time finding well trained and educated organists. So while sometimes I feel like a member of a dying breed, on the other hand, I can piece together enough money to get by just by working as a musician. Now don't get the wrong idea, I'm not getting rich, but I can live. For many years I was pretty poor because it was too difficult to get a well paying church job.
I don't know what the jobs are like in Denmark, but I can say that if you were here in the USA, and you asked me about becoming an organist, my advice would be that it is a difficult life. You should ask yourself if you could see yourself doing anything else. If the answer is yes, do the other thing. Those of us who make a living as organists (here, anyhow) are mostly people who just can't see ourselves doing anything else. It takes a lot of determination, patience, and practice to become good enough to compete successfully for a well paying job.
Besides that, the organ itself is in danger of becoming a rare instrument because of the proliferation of electronic substitute instruments. It takes strong willed and determined (and well educated) people to fight to keep our instrument alive. There are still lots of pipe organs, but electronic music, computers, etc have brought about a group of people who don't even understand why the organ might be an important instrument. So our job as organists is more difficult than it used to be.
I don't know if this advice is helpful for you in Denmark, but this is the advice I would give here.
Thomas Dressler