Limit Break needs to be changed

[Zaft]

The problem is that you need 4 stacks of whatever the debuff is called, so you have to hold the two snakes for a long period of time, which in turn results in less overall damage to the big snake (compared to having a level 3 limit break).
No matter how high your DPS is, you aren't going to down the big snake before Twintania drops with i90 gear, unless you have the debuff stacked.

You might be killing the second Hygieia too soon. Here's what we do:

Kill one Hygieia, take the second one to low HP, DPS Asclepius until next set of divebombs begin, kill second Hygieia after 1st divebomb, finish divebombs, LB2, finish off the Hygieias, finish off Asclepius. We often have a good 10+ seconds before Twintania lands doing this method.

Again though, raiding is about finding what works for your group. Experiment a little and see what you can do to improve.