If you don't want to run that command, you can try running EasyApache to rebuild Apache/PHP with correct mysql client extension, although that was already run probably, if you choose the unattended upgrade, as some of your sites wouldn't work after the MySQL upgrade if it wasn't done.
Also, I don't think you will have any issues if you run
/usr/local/cpanel/scripts/check_cpanel_rpms --fix.
I have run the command multiple times on servers to resolve issues just like this, and have never had issues with it.
If you have made a MySQL upgrade through WHM, the command should just remove old MySQL RPMs and leave the current MySQL 5.6 RPMs in place.
You should backup all your databases before running the command just to be safe, in case anything goes wrong, but it shouldn't cause any MySQL downtime.
You can backup all databases by running
mysqldump --all-databases > all_databases.sql to backup all databases in one file, or
for i in `mysql -e "show databases;"`; do mysqldump $i > /folder_path/$i.sql; done to backup each database in its own file.