Anything you do with the trial version will include a watermark... that is why it is FREE for 30 days
The watermark does not, as far as I know, go away when you pay and enter the code... you are not supposed to do real work with a trial version
No, the watermark is part of the trial. This FAQ Entry will tell you what to do to Export a Project without it. You will need to both buy the program, and then follow the exact steps in the FAQ entry.