Correct, the database saves only player, league, and season information. Stats are not saved, but computed on demand from your saved tournaments. Make sure you save each tournament that you run as a unique file (don't save your next tournament using the previous tournament's filename, for example). It's OK to load a previously saved tournament and Reset it to use it as a convenient starting point for your next tournament, just make sure you save it ("Save As...") with a new name.
The TD doesn't consider your tournament "over" until only one player remains. Are you physically ending your tournaments before that happens?