RCI does furnish chairs at Labadee. If you go to the beaches in Cayman and want a chair, go to the Holiday Inn and pay $10 and you can have one with a nice beach to go with it. I tried a few of of the others and like HI best.
In Ocho RIos, I would suggest the same. I usually head to Jamaica Grande for the day and for a nominal fee you get a chair, a decent beach, nice pool and plenty of availability for food and drink.