If I'm to draw and discard X cards from the Encounter deck and it runs out, am I to shuffle the discard pile and continue drawing/discarding?

My thought is, if it's during the Quest phase, then yes. Otherwise, no... but I'm not sure.

As you say -- if it's the quest phase, you would reshuffle and then complete revealing cards. If it's not the quest phase, you would only reshuffle if the card effect instructs you to do so. Otherwise, you reveal as many cards as you can without reshuffling, then stop.

Even during the quest phase, make sure to read the rules whenever there's a new mechanic, though, e.g., Sailing tests switch things up a little.

Quote

If the encounter deck does not contain enough cards to look at [for a Sailing test], shuffle the encounter discard pile back into the encounter deck first .

As one exception, in the event that an effect tells me to dicard from the encounter deck until a certain type of card is discarded and no such card exists in the encounter deck any more, I always figure the fairest thing to do is to discard the whole deck, then reshuffle it and stop there rather than be stuck in an infinite loop of discarding and reshuffling.