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. This is not strictly an Antigua resort but we have included it as for the lack of lower end accomodations.
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.