What version of Premiere Elements, what computer operating system?
You can remove the Adobe watermark from the tryout project file only when it is opened in the purchased product. At that time, you can do either of two things
a. Timeline Menu/Delete Rendered Files and Timeline render the content
b. Do nothing...some of us have found that just opening the tryout project file in the purchased product removes the Adobe watermark
You would need to see which, "a" or "b", works for you.
But, here is the part that I believe applies to your situation as you have described it....
If you export a file or burn to disc while using the tryout, you cannot remove the Adobe watermark from those exports even after you have purchased the product.