I actually just got to beating these two guys last night. The battles were quite tough and long for my party. I ended up using the same strategy for both. My party consists of a Warrior, Monk, White Mage, and Black Mage. My guys are around the level 52-54 range. In each battle the Monk could not sustain the damage very well and would die quite easily. After a few tries, I found it more economical to leave him dead. The party is more efficient without him.

At the start of each battle, I immediately used the tonic to increase HPs temporarily (giant tonic maybe?). Anyway, I made sure my White and Black Mages were close to 999. I also made sure my White Mage was upped to 999 MP as she would be using the magic in this battle. My other initial task was to cast Invis on the party and maybe even Protect if I remember correctly.

Once I was set with my 3 alive party members, my basic strategy was this:
Knight - Would cast things like ether (on White Mage) and potions depending on the situation.
White Mage - Would almost always cast Healaga on the party which would replenish probably around 250 HP per person. Unless she had another pressing issue (like casting Full Life on one of the other two members), I would cast Healaga religiously.
Black Mage - Would only do one thing: Use his judgement staff to cast Flare. It took about 550 a shot, and he was the only member of mine taking damage from the enemy.

The bosses always strike first, so I found important to be proactive and cast Healaga with the White Mage even if my party was full (unless she needed to revive a dead member) because the boss would take his damage at the start of a turn, and then she would heal up the party immediately afterwards. I needed to use one or maybe two dry ethers in the course of the battle as well as numerous regular ethers.

I found the whole key to the Shinryu battle was sustaining his damage long enough for him to run out of MP. I don't know what his deal is, but eventually he will stop casting Flare (which is the most brutal for the party), and I suspect it's because he is out of MP. The battle is much more controllable at that point, and none of my party members died after he stopped that attack.

Basically, the battles did take a long time (over a half-hour each I think), as my strategy was only to pick away at him with the judgement staff and make sure my guys were healthy. It'll probably go more quickly if you can use 4 members, but in my case, the Monk was just a burden to the team because he died too frequently and didn't have any kind of useful attack or function for the party. The time spent to revive him ends up being better spent healing the rest of the alive party.