From your tagged islands look into DR and Jamaica. Or Mexico..

You won't find good and cheap AI in the Bahamas or Puerto Rico

The Caribbean coast of Mexico has the most inexpensive AI options (into Cancun then south along the 'Mayan Riviera'). Much more so than Caribbean islands.

In Jamaica you can look at Negril on the 7-mile beach but I'm not sure any are quite that cheap (only $70 per day per person is pretty low for something 'good').

How long will you want to stay? Is that $1000 each or for both together? Does that budget include your air fare?

I have stayed at many "cheap" AI's in Mexico. Do realize though that cheap does not = good. We stay at those places for the breakfast buffet and drinks only. We go out for lunch and dinner, so we can get the best of both worlds. Since this is an honeymoon, do you really want to go "cheap"?

The most inexpensive AI will be found in Cozumel, MX. Look at the El Cozumeleno. With air and AI hotel for 2 out of Texas I can do a cheap AI for around $1,300 for 4 nights, but I know what I am getting into and my expectations are set accordingly.

<Is that $1000 each or for both together?>

Both, the OP said: "Looking to keep the place around $1000 for the two of us."

What they don't say is how many days they are hoping for for that price. If it's a long weekend, maybe. If it's a week or more, I don't think that will get you much, if anything in the Caribbean. Sorry to say.

Ya, its going to be a week hopefully and the $1000 would not include airfare. So $1000-$1200 for both people for a week.

And can you give specific places that would have these rates?

You are going to have trouble finding any "good" all inclusive resort for only $500 per person for a week. You might be better off looking for a small hotel or condo.

I suggest that you check some of the on-lime booking sites and see what you can get for what you want to spend and then come back and ask more specific questions if needed.

Here is a list of AI hotels in Mexico:

http://www.mexicoallinclusive.com/mexico_all-inclusive_resorts.cfm


Here is a list of AI hotels in the Caribbean:

http://www.caribbeanallinclusive.com/caribbean_all_inclusive_resorts.cfm