We stayed at Belizean Shores (north of the cut) from 8/18 to 8/28.
There were not many mosquitos in the resort, but plenty of them on the Golf Cart road. Especially in the evenings, stay away from the road and swimming pool.
In general, if there is no wind, there are mosquitos.
I kept thinking: how do locals deal with mosquitos ?
Other than, we really enjoyed AC very much. Resort staff and (most) locals are very friendly. Sharks and rays are such a life-time experience. We ate so much lobster.
Enjoy your stay there.