This problem is still occurring for me. It appears to be a OSX Mavericks related problem Flash Builder inherited from eclipse indigo - Bug 420682 – In Indigo, Open File dialog only appears once in OS X Mavericks (10.9)
Disappointing that Adobe hasn't chimed in on this. The problem was fixed months ago and they probably just need to distribute some updated eclipse bundles to patch it. Adobe, are you here?
I've got the same problem.
As you said, the only workaround is to copy the path from somewhere else and paste it into the text field, thus avoiding the file chooser from opening up altogether. I really hope this gets fixed quickly, though. Very annoying.