Dana1,

If you do a search you will find similar questions with a lot of great suggestions.

I will mention a few

Ivanhoe in town a/c no pool
Bayview a proper hotel with a/c , pool restaurant
Winnifred Breeze guest house type set up but private entrance Mrs. Scott is the owner 876-388-1129
Tim Bamboo a proper hotel no pool
Match Resort proper hotel
I have either toured or stayed at the above mention properties let me know if you have questions, most hotels are not within walking distance to water. Maybe Winnifred Breeze to Winifred Beach
Frenchman Cove is on the beach but I don't think it's in the budget you mentioned, but I could be wrong. The beach there is lovely

I have used AirBNB with good success also check tripadvisor there are a number of properties listed that I have not mentioned