When I was my guilds main healer, my guild leader used to chain pull all the time. I was under geared at first (doing upwards of +12 keys, with an ilvl that was wayyy too low for that…) and we managed to time the keys still. It’s stressful, yes, but if the tank is worth his weight in salt, he will pause in pulls long enough for you to sit down to start drinking. Once you get into the swing of it, it’s pretty fun. Until you encounter squishy tanks who expect to be able to pull like that, then die almost instantly and try to shift the blame to you… Which I have found usually works.
However, as people have stated since before this post, the name of the game in mythic + dungeons is to go fast. As a healer, you need to get used to the stress of it, and learn when you can stop for a quick drink. ~60% of mana should be enough to get you through a few packs of mobs, as well as a boss. Definitely don’t take it to heart when people say you’re a bad healer, though. I’d strongly suggest finding a tank that runs dungeons like that and trying to stick with him/her, because in my opinion that’s the best way to learn.