Everything has 2 sites and of course not everyone is a scammer trying to get money out of international students. I'm just giving you my version of the situation
Signing up as many places you can is your best bet, and then just wait. Wait and hope - search the private market but just be careful. Ask your university if they have any ideas, their accommodation office is bound to have some contact on that market at least. Contacts they don't usually give out unless people ask.
The thing about "building more dorms" is that a lot of young danish people are freaking picky. We don't wanna live in dorms unless we really have too, people are going for 1-room apartments with your bath and you own kitchen rather than having to share with anyone. Building more dorms would just mean a lot of empty rooms.