If you are not staying for long or a standard week (where it is hard to beat all inclusive deals), then you might look at this eco-friendly accommodation option.
About Dickenson Bay Cottages
It is spacious and has a pool, albeit not a massive one, but it is certainly okay for the size of the resort. You could classify this as a boutique hotel perhaps - Antigua style.
How far is this away from Dickenson Beach? It is not on the beach is the first thing to mention, but a 10 minutes walk and you are there.