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 ) (just use the same apple account)

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 )