I think it would be best to play more than one.
I don't think playing a shield healer will do all that much as preperation because shields are pretty much an afterthought in this game and nothing a healer truly relies on like in other games where shields are by far the most mana and time efficient while direct heals fall short and are expensive emergency heals or where regens are more like a weak but steady baseline healing.
We'll also hopefully finally get a big overhaul on all healers which means getting too comfortable to the point of being stuck in a certain playstyle will do more harm than good. But even though an overhaul is probably coming, playing different healers to get a general feel for what they can and can't do aswell right now aswell as their unique "rhythm" will help you become a much better healer.
For DF content it won't really matter that much but if you want to get a more in-depths feeling for it, at least leveling all of them with lots of DF is the way to go. You have plenty of time until EW.
SCH and AST have an equally expansive toolkit but they use it very differently. AST can simply keep oGCDs rolling without a downside while many SCH oGCDs are tied to AF as an additional resource. WHM has a very barebone toolkit and it relies on making the most of your limited resources and doing things in the right order to not waste potency and mobility.
Playing one healer (Sage in your case come EW) well is only half of being a good healer, having an understanding for all of them is the other half. And nothing teaches the latter like personal experience.