Is your phone rooted? If it is, you can find the Electrum wallet files in the /data/data/org.electrum.electrum directory and manually extract them/back them up from there.
If it isn't rooted, then you should be able to back up the entire directory using the Android Software Development Kit with your phone connected to your computer. Have a look at this Stack Overflow post for more instructions:
https://stackoverflow.com/questions/13006315/how-to-access-data-data-folder-in-android-device. However, I don't think you will be able to write your backed up wallet files back to your phone after your fresh install without root access.
Have you tried uninstalling the different version on the different user account and just running the original version again?