Yes nbssn is right about installing apk over apk, I forgot about different signature, but there is module for XPOSED that dissable this check, but on unrooted phone it is not possible, maybe try his opinion but I suppose that you don't have installed custom recovery because if yes root won't be problem..... Maybe try to backup your app settings via Google Acount (I don't kniw how, but I thonk it 's possible) then uninstall app install original and recover data from account, I have never done this opinion, but somebody tells me that when he performed restore system opinion, that after this everything was recovered from his account...

There's only a few ways this can possibly be done...

If you have a Custom Recovery like TWRP, CWM, etc... You can try the following app called ZIPme from the Play Store and create a flashable zip of the App's "Data Only" and uninstall the modified App then reinstall the unmodified app and then flash the zip file in your Custom Recovery to recover the Data.

https://play.google.com/store/apps/details?id=zip.me

If that's not an option...

There's always the following 2 App options to perform the same type of backup of the App's Data like the above steps but, after installing the unmodified app, use what you had backed up from the following App options to restore the App's data.

https://play.google.com/store/apps/details?id=com.koushikdutta.backup

https://f-droid.org/packages/dk.jens.backup/

I wish you the best of luck!

~Snausages

Basically no, you cant install an app over another app without root if the signatures do not match.

And yes you cant go online because you have rebuild the app with a different signature. Because now google play games check the signature or asks the app to check it. If it's messed up it won't let you sign in.  

Try to find unmodified apk somewhere... Download and install