I would really recommend Healbot it frees up keybinds for cds, CC and some damage spells.
If you dont like healbot there is Vuhdo which you can try. Place this at the bottom middle of your screen, this will keep your screen open to observe the fight.
Grid could also work for you, but a healing addon is a must there are a few exceptions to this rule but they are far and few.
Now the signs of a good healer is one that can keep an eye on the raid and heal them while staying out of the fire also being able to determine what heal would be most efficient in the current damage taken, Who must he prioritize and who is going to be a waste of mana.
When lots of mobs approach trust your group each has the assigned tasks, The Tank should pick them all up if he has aggro on all you should throw CDs on him, he himself should throw out a few. The dps themselves should pop a few cds if it raid wide damage use your raid wide cooldowns eg. barrier or Hym.
It takes practice to do this and like one of the people before me said get a guild not Pugs. I have been healing for 3 years and I still do sometimes panic but it has become far less like once in a blue moon when a new situation comes around that I havent faced before.
It just takes time
P.s. A good way to tell if someone is standing in stuff they shouldnt, by just looking at their health bars they will be taking a lot more damage