Do you only get one chance at casting a spell?
Yes, you get only once chance.
The key phrase is &quot;cast and exhaust&quot;. An exhausted card can't be used again until it's un-exhausted during the next turn's upkeep. And the spell is exhausted whether or not you succeed at casting it.
The rules are a little bit fuzzy on this, but this official FAQ has a pretty clear answer:

Does an unsuccessful attempt to cast a spell still exhaust it?
Yes.