You don't have to stay at the airport after you check in and check your bags, no. As far as how EARLY you can check in AND check your bags, it depends on the airline and what day of the week it is. Many or most won't let you check in and check bags more than 4 hours before, although AA seems to have a lot more leeway about letting you check in earlier and leaving your bags than other airlines do. But if you're wanting to go say, three hours before your flight, hopefully you shouldn't have a problem checking your bags. However, some people recently reported Jet Blue wasn't let people check bags until two hours before. Best probably if you have time on island is to drop by the airport and ask in person.