I wrote only a little while ago about how different I found healing on my Paladin to be as compared to my Druid. I've done a lot more healing since then, and I've found it to slowly start losing its stress as I became more competent. I'm still shaky and not completely confident about my skill as a Paladin healer, and Retribution is still my first love on Bellbell, but I've figured out some tips for those who have new alts who do the same job as their main.
It's important to remember, first of all, that even if the goal of your job hasn't changed, a lot about how you go about achieving the goal has. You are not playing your main, so using the same style will not work. The sooner you can break out of the mindset of your main, the sooner you can begin shaping a new one for your alt. For example, when playing my Restoration Druid, I know that when I have HoTs on a character, they will continue to heal, whether I take some time to focus on another character or if I use more healing moves on them. However, with a Holy Paladin, what I heal is what they get. When a Flash of Light lands, the amount that is healed is the amount that is healed, and there is no continuation of healing, no ticks. When the group/raid is taking damage, I cannot just hit Wild Growth and spot some here and there. I have to immediately begin throwing out concentrated, focused heals.
It's also good to remember that you are not going to be the best immediately (barring some exceptions). You've spent years perfecting your Rogue's rotation, adapting to subtle changes, optimizing main hand and off hand, tweaking your talent tree and switching poisons. Moving over to a Fury Warrior, you may not have the rotation down. Your gear will probably be weaker. Be prepared to be slower at adapting to changes, of learning new ways to use moves or different ways to spec talents. Take it slow, take it easy, and let yourself make mistakes.
Don't jump right into the harder instances. You'll be kicking yourself and hating the alt if you go right to heroics when you should be doing 80 normal instances. Your main may have been able to do those instances in worse gear, but, remember, you've known that character for a long time. You've possibly been through several expansions with it. This alt has either been a long time in levelling, or is brand new. Or, perhaps, it's a dual spec you've never given a shot. Don't jump in head-first; use some water wings first.
And, goodness, if it's stressing you out, take a break. Have fun with it.