Best advice for a healer: remember that a tank should know what they are capable of pulling without having to lean hard on the healer to keep them alive. If they're going to pull everything leading up to the boss, don't stop to try to heal them (or the DPS for that matter). If you have an ability that is instant cast available, use that as you run behind the tank if they are properly generating enmity on everything. Otherwise, wait until the tank finally stops so you can also stop and start casting your heals.

If the tank dies because they pulled more than they can handle with a new healer in the group, that's on them. Once you have Swiftcast, you can use that with your resurrection ability and get the tank back up immediately while healing the other party members now taking damage. That's usually enough to keep the entire party from dying. Most tanks will usually figure out at that point that they need to slow down a bit.

On the flip side: look at this as a challenge to become the best healer you can be. At lower levels, it's mostly out of your control because of the need to stop moving to cast. As you start gaining instant cast abilities, especially the ones off the global cooldown, you will be better able to handle the chain pulls as a healer.

If you continue to remain uncomfortable as a healer, that's okay. Maybe you'll enjoy a DPS or even tank job more than the healing jobs. I've got friends who are fine healing content up through level 60-70 but prefer to let me do the healing in the higher level content.