Actually, Llund.. .. . you are partially correct, but not 100%. .
Kingston / Montego Bay are the only international airports. There are two smaller airports that I have flown into FROM Montego Bay though. One is in Negril (directly across the street from Couples) and another one in Boscobel Bay. One of the charter airlines (TimAir) flies to both of them, and I have used them extensively over the years.
I will also agree that Air Jamaica is the absolute best for Int'l flights.. ..Their inter-island flights? Blech. I will never take any airline BUT Air Jamaica to the island, but I will never take one of their puddle jumpers again

Heck, I am not even sure if they still have them.