NO ...no SYNC needed, u need to download the latest version of the app ON PC so it goes in the cloud and doesn't show as update when u acces appstore on iphone ...( this will make 1 badge disapear ) 
the other badge is from the app allready installed on the phone ... and there is no way to remove that without deleting iTunesMetadata.plist (i think u can acces the phone ROOT FILESYSTEM even without jailbreak - check youtube) ( anyway why do u keep using iOs 6 if u don't jailbreak ... i don't see the point )