I would go on cheapcaribbean.com - I booked a 6 day, all inclusive trip to mexico from there for under $900, including airfare. They have a bunch of countries and hotel options to choose from. 

I am not sure how cheap you want it. But prices along the coast of South Carolina are generally quite low in December and the weather is very temperate.  Check winter specials in Charleston, Myrtle Beach, Hilton Head and Beaufort.

Biloxi, MS. 

One hour to New Orleans. Beautiful coastline. Exciting night life. One hour from Mobile. Modern Art Museum. Fun to walk around. Many beautiful Casinos.

I like Russel's ideas and would add Savannah and perhaps St Augustine. Another Christmas place is Asheville where the Biltmore Estate puts on quite a show.

It's not very south, but the Gaylord Opryland Resort in Nashville is a cool place at Christmas time, and it's a neat town. If you want warm, I just took a cruise that week from Miami. We had great weather. Christmas week can be expensive but a week ahead or after makes a HUGE difference. Share an inside cabin to keep the cost low. 

South Carolina would be a great choice, I would go farther South into Florida the west side a little planning and booking should help keep the budget in check

Cuba is often cheep, take an all inclusive package.

Bring a tent and camp along the coast of Mexico.  Research before the nicest campground at the nicest beach and stay there.  Eat local fare and just vibe on the sleepy fishing port culture.  You could easily stretch your dollars, by camping (it can be romantic if your adventuresome!) and do day trips if you get bored with the beach scene.  

How about Niagara-on-the-Lake for some Ice Wine? :^)