Azala and the Black Tyranno are a dangerous threat indeed. Fortunately, they can be taken down with the right strategy.
Your first focus is to take out Azala for it'll use moves that can cause status problems as well as dealing damage to your party. Use either Single or Dual Techs to take out Azala but when it goes, it'll use Azala's Wrath which will cause Sap to all members.
Once that is done, concentrate on the Black Tyrano but make sure to heal in the meantime. When it lowers its defense in hopes of gathering power, take it into precaution. In the meantime, use various Dual Techs such as Thunder Chomp, Ice Toss and Ice Sword to dish out major damage. It'll roar five times signaling a countdown so ensure that you attack and heal when necessary. After the fifth roar, it'll unleash a powerful flame attack attacking all members for serious damage but if all your party members have Ruby Vests, the damage will be cut in half. After that, attack with your strongest abilities and the Black Tyranno will go down.
What happens afterwards is something I wish not to disclose... watch for yourself what I mean.