As others have said, "festive season" is the most expensive time of the year on Anguilla and places that would be in your budget a week before and a week after are not during the holidays.
Reading between the lines I sense that your plan is to NOT be home during that time, so moving the trip may not be an option or may miss the point of the exercise.
I am sure folks can recommend places more off the beaten path that would work for accommodation price wise, but the location would require a car or many expensive taxi trips, even if you hit the supermarket and cooked in a lot, you'd still want to get to the beach and distance would likely be a problem due to the high cost of taxis.
If the point is to be away over the holidays, perhaps Mexico might work and still give you warmth and easy access to food and the beach all in one location?