This is the exact path I followed, MNK and then SCH. They're still my two main jobs. I play solely on controller.
The two most important things to understand about SCH are that you're not simply a pet class, and that you're not simply a healer.
Your fairy is less of a primary thing for you than it is an accessory. She is, functionally, your stance, as you have two options with meaningful differences, but you are still functionally the same in either one, each just adds some extra stuff.
The attitude of a SCH is so far removed from what it is to be "just a healer." Though a non-negligible amount of what made SCH special was removed as of Stormblood, the general play style remains the same; Think of yourself as a support character, rather than a straight healer: You proactively prevent damage, and prepare the party for the situations you're aware that they're going to get into, in order to minimize the damage actually taken by them. Your goal is to leverage your shielding, mitigation abilities and fairy in conjunction to buy yourself as much time to flex as possible - be it through healing multiple people that need it, or laying down DPS on enemies whenever you can.
In essence, stay a step ahead of your teammates, get them prepped for what they're going to run into (usually AOEs...) and then do what needs to be done from there.
Though our ability to do so has been hampered and the job isn't in the best spot right now, it's still functional and still has the same general play style - just, you'll be casting Physick a lot more often than you would previously.
As a general play style, healing is still "heal people that need to be healed, and do damage when they don't," for all three healers, WHM is very blunt and straight-forward about it, having the most up-front and powerful direct heals and damage spells of the three. SCH is, thematically and functionally, a job about tactically-useful spells and abilities that feels rewarding to play well. AST is like a cross between the two, emphasizing supporting the party's offensive output over their own damage, though they're subject to the randomness of card drawing.
8 vulnerability stacks and I provoked the enemy because I'm dumb.