Do I need full recovery mode when I have multiple daily backup?
Setting a database to full recovery is not enough by itself to ensure you have point in time recoverability. You also need to take regular log backups. You should also be taking time regularly to test that you personally know how to execute the required commands to actually perform a restore to an arbitrary point-in-time. 

In lieu of running loads of home brewed full backups, the fire-and-forget solution you want is probably to just install Ola Hallengren's backup solution. Running this script installs the required objects and deploys the required jobs to manage all of the full recovery databases on an instance. You will find these jobs 1 under SQL Server Agent > Jobs named by default DatabaseBackup - USER_DATABASES ... and similar. Add a schedule to these jobs that suits you 2 and disable other backup jobs to prevent conflicts. 

You should still need to regularly take time in the normal course of business to ensure you and others on your team are able to actually restore the backups you have, regardless of how you take them. 



1: assuming you're using SSMS here

2: 5 times daily if you like, but probably a bit less is fine