You need to "keep the story moving"? Isn't the party breaking down and potentially splitting up part of keeping the story moving? (See also: the fellowship's disintegration in the Lord of the Rings.)
However, that's just glib. Many RPGs, both as written and as played, encourage the group to work together to solve a given problem. In, say, D&D, splitting up the party tends to be a recipe for a TPK. And while some people think that's okay, it fails to match up with the fiction that inspired D&D and many of it's players. A single character being obstinate can lead to an un-fun stalemate. So what do you do?
Avoid it in the first place. If you're playing this sort of game, everyone should build characters willing to work together. A lot of these problems are caused by someone creating a character with the "I would never" blockages built in from minute one. I've done it myself, because it seems like a cool idea that you see in fiction all the time. However, the needs of fiction and a party-based RPG are very different.
Metagame. If you feel trapped by your character and are causing the stalemate, just be blunt. "Playing true to my character will just lead to not-fun for everyone. So I'm going to quietly ignore it this time so we can keep going." Metagaming isn't as terrible as people make it out to be. If metagaming is the bridge between not-fun and fun, cross it!
Ask your GM what he's thinking. The GM tends to have an idea of a solution. If you've reached the land of not-fun, the fastest route out might be to get some advice. In your example, presumably your GM created the effect causing the problem as well as the situation requiring going into the mists. I doubt his plan is, "Aha, now they'll spend the next few sessions arguing over it, and I can catch up on my cat videos on YouTube!" He thinks it's a solvable problem. The problem may be as simple as your overlooking something that to the GM seems painfully obvious. (As a GM, this happens all the time.) You might only need a small prompt ("Hey, do you guys remember that church you visited two months ago. The one whose high priest specialized in curses?") or a big one ("You can get the curse lifted, go see High Priest Bob.").