Late Sept- Jan Inquiry

My fiancé and I are getting married in late September! We're opening to going on our honeymoon from that date or up until January since hurricane season is prominent during September.

We have a budget of $5k and would fly out of LAX. Any ideas of where we should go ? Both my fiancé and I care about great food & need a healthy mix of adventure/relaxing. I was thinking about Costa Rica originally.

Thanks in advance!