I've tried other hotels, but I always go back to Ruby's. The oceanfront rooms are US$50 and are right on the sand. They're hurricane-proof, concrete construction, so noise hasn't been a problem.
As to Belize City, a good bet is the Best Western. It's out of downtown a little, so it's removed from the hustle of the city center. I would recommend using the city as a base for mainland excursions, because staying on Ambergris adds time and expense to your mainland travels.
I export Traveller's rum, and business requires that I split my time in Belize between the Cayes and the City. If business is going to keep me in Belize City for a few days, I stay on the mainland overnight. The alternate is staying in San Pedro Town and hopping back and forth to the City. So, I'd either spend a couple of hours every day on the water taxi or spend a lot of money on airfare. Your mileage, as they say, may vary.
Have a good time.