Do not panic.
What did you purchase - Premiere Elements 14 as a download from Adobe or something else from somewhere else?
Be that as it may, when you open the program to the workspace are you getting one of those tryout messages when you are using the tryout?
If so, you should see a Continue tryout or at the least a license this product tab on the message. Click on that license this product and follow the instructions. Have your purchased serial number at hand in case it is asked for.
Also, when you go to close your project/program, do not go to Help Menu/Sign Out. That deactivates the program. Just use File Menu/Exit or click on the X close button and then respond to save project or not.
After reading the above, please let us know if everything is under control. If not, let us know, and we will work on this some more.