Cutter's Cry is split into three distinct visual styles, each of which contain different types of enemies, different environmental effects, and are separated by a boss.
The Dry Sands consists of two single-room chambers and series of tunnels connecting to a slightly large chamber containing the first boss. In order to progress, you will need to kill 8 enemies in the first two chambers to spawn a Shifting Sands, which will progress you to the next chamber. Throughout this area, you will encounter sand traps, which will deal ~100-500 damage every few seconds while you are standing on them.
The second area, The Wet Sands, is a set of three chambers. Unlike the previous section, you do not have to kill anything to spawn the Shifting Sands. These are usually on the opposite side of the oasis from the position you start at.
For the first half of the fight, the Myrmidon Princess will attempt to hit the tank with Mandible Bite, which is easily dodged. She will also cast Silence on a random party member. If it lands on the healer, there isn't much to do other than wait it out. Try to keep the tanks HP high so they will not die during this.
Around 50%, the Myrmidon Princess will emote "The myrmidon princess senses danger and summons an army of myrmidon marshals." At this point, she will continually summon more adds for the rest of the fight. The Myrmidon Marshals are the highest priority to kill. These will attempt to heal the boss with Formic Pheromones, which should be stunned. The marshals will also use Mandible Bite, creating more ground indicators to dodge. Next are the Myrmidon Guards. These will use Mandible Bite, but will not use the heal.
Kill priority for this phase will be Myrmidon Marshals > Myrmidon Guards > Myrmidon Soldiers > Myrmidon Princess. If you have strong AoE, you can ignore the soldiers. Once the boss dies, take the Shifting Sands to The Wet Sands.
Alternative Strategy: Ignore Adds
If your healer is comfortable, you can ignore all the adds, allowing your healer to draw heal aggro. Then just kite them while the boss is burned down. While this is tested with a WHM healer, I am not sure how well it would work with a SCH due to lack of Regen.
Special thanks to Knight__Artorias for this tip!
Position the Giant Tunnel Worm so only the tank is in front of it to minimize Sandstorm damage. Sand Cyclone will occasionally be cast on a random player, dealing some damage and inflicting Sludge. Either remove Sludge or heal through it.
At 75% and 50%, the tunnel worm will burrow underground and select a random target. You will briefly get a red aggro arrow to indicate which player is being targeted. Sand Pillars will start to rapidly spawn in a line from a random part of the room to the location the player was standing. Once it reaches that location, it will use Earthbreak, erupting from the ground. If you are quick, you can run as soon as you are selected and dodge this.
At 25%, it will burrow again. Instead of selecting a target, however, it will cast Bottomless Desert, pulling everyone into the middle of the room. You will have a few seconds to quickly retreat from the middle to avoid Earthbreak.
On the pull, Chimera should be faced away from the group. This allows the tank to dance left and right, dodging Ram's Breath and Dragon's Breath without getting anyone else hit. Later in the fight, Chimera will cast Cacophony, which places a small targeting symbol on a random player and sends an orb at them. Once the orb reaches the target, it will stop moving and explode a few seconds later. Let the orb reach you and then everyone should move away from it. The orb appears to have an aggro table, so will go after the healer most of the time. If you're healing, just be aware you'll need to move from it frequently. You should also dodge Ram's Keeper, a circular AoE placed on a random party member.
And finally, the abilities that make this boss quite challenging. While dodging the previous abilities, the most important task for you to do is watch for the boss to emote either "The ram's eyes now burn blue" or "The dragon's eyes now burn violet." If he emotes "The ram's eyes now burn blue," everyone needs to run away from the boss, as he is about to unleash Ram's Voice. Likewise, if he emotes "The dragon's eyes now burn violet," everyone needs to run into melee range of the boss to avoid Dragon's Voice. This sounds simple, but when trying to avoid Ram's Keeper and the orb, you may not always notice the emote. Both of these abilities also have a cast bar, if you happen to miss the emote.
To put it simply, run away from the boss when there are blue eyes (Ram's Voice), and stack on the boss when there are violet eyes (Dragon's Voice).