It's possible that you can open version 11 project files in version 13 -- but doing so could corrupt your project files. (It certainly did happen in older versions of the program.)
So I would recommend finishing any projects in the same version of the program you started them in.
Or, if you do try to open an older project in a newer version of the program, save a copy of your project file. Just in case doing so corrupts the project.
In fact, as a rule of thumb, don't do it. It might work but it might not either.
Given that there are no guarantees about opening a project file created in an older version in a new version....for that I would ordinarily include work from a copy of the older version's project file. Classically once an older project file is opened and edited in a newer version, you cannot go back.
But, your question targets your concern about the Adobe watermark in the Premiere Elements 11 project file being opened in purchased Premiere Elements 13/13.1. That is a specialized issue. I have never tried that one version to another. Even in the scenario when only one version is involved, that may or may not be complicated. The classical story is that you use Timeline Menu/Delete Rendered Files in the purchased program to get rid of the Adobe watermark in the tryout project file opened in it. But, some of us have found that just opening the tryout project file in the purchased program automatically gets rid of the Adobe watermark. And, even files in the tryout project files that have Adobe watermark and no rendered files, nonetheless can present in the purchased program without Adobe watermark. Please refer to my recent study of this situation ATR Premiere Elements Troubleshooting: PE: Adobe Watermark Removal From Tryout
Bottom line on this part: no guarantees.
If possible, I would suggest that you leave the Premiere Elements 11 tryout installed when you are working with the purchased Premiere Elements 13/13.1 until you work out the situation.
Any questions, please ask.
Add On..As I see it, you cannot use the suggestion to finish an 11 project in 11 since the Adobe watermark will exist in the tryout 11 project file until you purchase the 11 program and use Timeline Menu/Delete Rendered Files or alternative that will remove the Adobe watermark.