American Airlines has round-trip flights from Austin and San Antonio (AUS, SAT) to Grand Cayman, Nassau, Bahamas, Providenciales, Turks & Caicos Islands, Belize, Punta Cana, Dominican Republic (GCM, NAS, GCM, PLS, BZE, PUJ) for $258-$380 with a connection. Most flights are between August to December. Very few weekends available. No Christmas dates available.
Note: Flights out of San Antonio are usually more expensive and and worse routing than Austin.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.