Bahahaha damn! x3

I so wish this was an actual Emote healers got.

I use White Mage in FF14, but this is the first I am hearing of Healer Adjust. What is it?

It is something said by a bad tank who over pulls or is reckless and doesn't use their mitigation as they should. People don't always use that exact phrase but the sentiment is the same. "Adjust to my inexperience/incompetence/recklessness."
It is basically passing the blame to the healer when things don't go well because of the tank's actions.
Luckily it has fallen off in use, but had a brief resurgence when a bunch of people came over from WoW, and some of the new players didn't want to take their time and learn the differences.
It is also used by reckless DPS players, who don't avoid the AOEs, because of their cast timers. Black Mages used to have a reputation for this, but not so much anymore.

Ah yes. I have felt the pain many times when leveling up my White Mage. True it took a bit to get the handle on how to properly use them, but when you do low level dungeons with Daily Roulettes and you get a stage where your good healing spells are gone and you are in a place you know hits tanks hard regardless, it doesn't help when said tank thinks over pulls is a constant good thing when you can't heal them fast enough. I actually rage quit a dungeon once because of the constant over pulls :P