Second question first: from the Visitjamaica site
Non-resident applications may be submitted anytime within the five (5) day period before the intended date of travel.
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Because the Antigen tests for antibodies and the vaccine creates antibodies in your body, if you take the Antigen test you will test positive. On the much more expensive PCR test you would probably test negative.
You have 3 choices, 1) wait on the vaccine, 2) take the more expensive PCR test and hope for the best, 3) contact the Visitjamaica.com site immediately and ask them what if any vaccine documentation and information you must have on hand to get past testing positive on the Antigen test.