I can definitely see adding the search bar to the add a post page. I'm not sure that having it forced upon people is the right approach. One thing that we've done is provide a FAQ for different forums that have has common questions show up over and over. This FAQ appears right inside the box where new posts are created to make it very obvious that people should check the FAQ before making a new thread.
My personal feeling is you want to make it obvious where people can get the answer to the question before they post but I don't find force to be a viable solution to any dilemma.
I agree in your perception of the "forced" part of this equation but thought I would mention it as a possibilty. I think that the more important idea is to differentiate from the standard search engine is ease of use offered at the most appropriate time.
Makes sense. The search bar on the threads, posts, forum, and now the add post pages are set to search the current forum. This, I believe gives people the opportunity you are looking for. The FAQ on some forums is an additional means to give people the opportunity to find what they need before posting.