Ok. Login to the new account and install the apps you want.
Optionally, to repair the old profile, you may try this app.
- Account Profile Fixer: https://www.carifred.com/apf/
Since you've backed up the data to the Public folder, the above app is worth a try. If it fixes the old profile, that would be great. Else, simply discard the old one and use the new account.