If you stay at an Arkham Adventure card with an event icon after failing it ...during your next turn, will you draw another event card before rolling dice?

From the bottom right of page 3 of the rules:

" Events

At the start of an investigator’s Resolution phase, if he is at an Adventure or Other World card with an event icon, he must draw one Event card and resolve it before resolving his Adventure card."

So if you are on an Adventure or Other World card during the start of your resolution phase you will need to draw an Event.

Yup, Devon nailed it.

Remember - this could create some confusion:

a) Entry effects are triggered at the end of the Movement Phase if you moved to an adventure featuring an Entry effect (but they do not trigger if you already are at that adventure), so, they are movement-related

b) Events are triggered at the beginning of the Resolution Phase, and they are not movement-related (i.e., whether you start on that adventure, or you just moved there, it doesn't matter, the Event is always resolved)