Seat sales come and go to Belize, as elsewhere. Recently, for example, AA was discounting fares drastically from Atlanta via its DFW hub, but not from most of its other markets. You just have to keep checking to see what fares are available on a specific date.
United I have not heard has any current plans to fly into Belize City. (It used to fly there, many years ago.) Delta has considered the market for several years, up to the point of measuring the runway at the international airport, but as of now is holding off on the service.