Grand Cayman doesn't have zip-lining but there are lots of activities, both natural and tourist venues. Sunset House is a nice option that includes boat diving, shore diving the house reef, airport transfers, breakfast, and some other expenses, see below.
Cayman isn't cheap, but there are reasonably-priced options for meals and activities and the Sunset House package includes a lot, and it is a good location for access to other island activities. Also, Cayman got lucky last year and didn't get hit by any hurricanes, so they are not currently rebuilding; and you won't have to worry about Zika or other tropical infections - if that is a concern.
Sunset House, Grand Cayman
https://www.sunsethouse.com
7-Night Dive Packages - Price includes daily 2 tank morning boat trips, unlimited shore diving on the afternoons of your boat diving days, round-trip airport transfers, daily fully made to order breakfast, welcome drink and gift and all tax and service charges.
2018 Summer Rates - April 1 - December 21, 2018
PP Double Occupancy, Courtyard Room: $1,598.65
You said "East Tennessee" so I am guessing that you would fly out of Knoxville or Chattanooga, right? I randomly picked a week in October and found RT airfares on American from TYS to GCM for $688 per person, with one stop in Charlotte, see screenshot below. There were other flight options with AA and other airlines for similar prices.
So now the total price is $4574 for air, transfers, hotel, diving, breakfast, taxes, and tips (except maybe dive shop tips?) leaving around $1500 in your budget for other expenses; how does that sound to you?
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