If you intend to marry in Jamaica and enter the USA as married persons, then the only practical method is Immigrant Visa Processing.
The U.S. government needs to "vet" your spouse, before an immigrant visa will be granted. This can take up to three-four months to accomplish. Obviously not a very happy proposition. However, let me suggest a "what if" for you.
Suppose that a Jamaican citizen were to come to visit the USA on a tourist visa with no intent of getting married to you or anyone else. And, let's just suppose you met this person by accident, fell in love and got married on impulse.
Under such circumstances, your new spouse could get an immigrant visa while waiting in the USA, and he would not be require to leave. He could also obtain an Employment Authorization Document (EAD).
So, maybe a beach wedding could be changed to a honeymoon on the beach, instead. Something to consider.
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