I think everyone who's run an instance and a raid has been in this situation -- you wait all night for a Karazhan run, it fills, so you decide to go run a heroic or some other five man, and then a few minutes in, you get a whisper: "Someone left, and we need a healer! Are you still in?"
Sigh. An embarrassment of riches. So now what do you do: ditch your newfound 5man group for the raid you originally wanted, or stick with the group you signed on with, passing up the original raid? Neither way is very smooth -- either you disappoint the folks you're with, and look like an elitist raider, or you tell the raiders that you're already in a group and run the risk that they won't ask you again. So awkward -- it's like choosing a social circle in high school all over again.
Usually, the best course of action I've found is to cover yourself before you leave either group -- jump into LFG or ping the guild for a replacement before you leave anybody behind. Short of finding someone to fill in for your fill in, the next best option is probably to explain yourself well and promise help next time -- hopefully the group you're with will understand that raiding is fun and/or the raiders will understand that you couldn't wait and keep you on the list for next time. Any other ideas for getting out of an instance pinch like this?