You're only at copperbell. Honestly, take a few hours and level a Conjurer or a tank class. Keep it on par with your favorite class so you can get into story instances instantly.
There are MANY more instanced fights to come in the storyline. Suck it up now, and you'll be helping a party of other people stuck in line while helping yourself. I have a lv50 DRG from 1.0 but I'm running through story on a GLD/PLD because it'd be hell getting into instances otherwise, plus I'll have a good role available when I unlock endgame again.

Don't just complain, DO something about it

edit:
I should say though, that I agree there need to be more classes that can queue as healer. Either that or the duty finder needs to be more flexible.