As per the Release and Upgrade notes as well as the information contained in the following sticky post at the top of the forum here, before importing the new TMP package, you should remove the previous version by deleting the “TextMesh Pro” folder.
Provided you are upgrading to and from the same version type, like source code to source code or DLL to DLL, this should work as expected.
Also make sure that you do end up with the correct version of the product for the version of Unity that you are using as there is still a bug where the Asset Store can end up providing the wrong version of the asset. If that is the case, I provided direct links to download those releases in that sticky thread / post I linked above.
Script references should all be maintained as the GUIDs and FileIDs for these version is consistent between releases.
Give this process another try. Again be sure to backup your project first and let me know if you need further assistance.