As an AST main who uses the DF roulette a lot I have learned to be very adaptable on the fly. Going in to DF you never know what you are going to get for Tank or DPS and what they are going to be like.
So this is what I do:
1) Skip opening cutscene UNLESS it is one I have not done many times before.
2) Immediately cast Protect as there is usually one or more who skipped the CS with me.
3) Those who are not watching the CS get Stone Skin even as i cycle my cards to begin to set up for the run.
4) This usually gives me enough time to get Protect and SS on the final member of our team.

Now, if everyone is in a hurry and not waiting for any of the shields or buffs I cycle my cards tap sprint to keep up. As the group engages the first mob I cast protect centered on myself and make sure I am positioned to get both the ranged and melee members of the team. I then switch over to applying a HoT and top up on the tank. I then either begin DPSing or healing dependent upon the skill of my party and their needs. AS the mobs die off and we get close to the end of the first group I switch to casting SS first on ranged and then melee and finish with the tank to time it just as the last mob dies. This means we are already buffed and ready for the next group without anyone having to stop as they can do their stuff without having to wait on me. I then reapply SS on those who take hits during the end of subsequent mob encounters. (And never stop cycling and applying cards to my team.)

Long story short... Adapt to your team and go with the flow. There are ways to keep up without stressing even at the early stages of levelling. It is our job as healers to be calm even when the worst possible event happens because our team relies on us to keep them alive and I will die trying to do just that.