If you're having trouble hitting the targets before the DPS, use Sprint to fash to the next pull before everyone else. Also I'm gonna go ahead and say this just in case... STOP TRYING TO PULL SHIT IN SWORD OATH/DELIVERANCE! Maybe this won't apply to you, but it will definitely apply to a bunch of moronic tanks I see out there.

Now I won't say I never have enmity issues, but there's an easy way to assess what's going on. If you're losing enmity to a group of enemies all at once, it means you aren't Flashing enough. If it's just 1 at a time though, it just means one of the DPS decided to focus something different than you. This happens regardless of what job they're playing, and you just gotta smack it with something (preferably your enmity attack) to get it back.

After some practice, you can usually tell which enemy you're about to lose enmity with based on how low its HP is and when the last time you hit it was.

Also, try doing big pulls to motivate the Red Mages to use their AoE attacks more. It's a lot easier to keep enmity from AoE attacks.