You are right the problem is NP-complete. So your best chance is to solve it by brute-force (if you find a polynomial algorithm please tell me, we can both be rich =)).
What I suggest you is to take a look at backtracking. It will allow you to write an elegant (and yet slow given your input size) solution to the crossword problem.
If you need more inspirational material take a look at this solver that uses backtracking as a method to navigate the solution tree.
Note that there are algorithms out there that might in practice perform better than a pure brute-force (even though still of exponential complexity).
Also, a quick search on scholar reveals a good number of papers on the topic that you might want to take a look at, such as the followings:
- using genetic algorithm
- using a probabilistic approach
