The program is still using the preview files that were created by the trial version.
Go to the Adobe Premiere Elements Preview Files folder that is located in the same directory folder as your project file and delete any files that are in it.
Where is the Adobe watermark retained - in a video exported from the tryout or in the project file from the tryout?
You cannot remove the Adobe watermark from an export from the tryout, be the export a file saved to the computer hard drive or a burn to disc product.
You can remove the Adobe watermark from the project file from the tryout by opening that project file in the purchased version and using Timeline/Delete Rendered Files...or, as already suggested, going to the Adobe Premiere Elements Previews Files Folder and deleting its content of rendered files.
Some of us have found that just opening the tryout from the tryout in the purchased version automatically results in the disappearance of the Adobe watermark.
In your situation as is, are all the Adobe watermarks present in the purchased program's project file or are most gone, not all?. I ask this because of an interesting recent case where a user was able to get rid of the Adobe watermark from the tryout's project file in the purchased product for all but the titles. And, that brought up the interesting point that, under most cases, titles do not have rendered files.
Please supply more details.