hi Tere,
Between Paradise Island and Nassau, I would recommend Nassau as the place to find cheaper accommodation.
Nassau offers a wide range of
budget hotels near the beach or in the downtown area. Although, without a doubt, downtown is your best bet for the absolute lowest prices.
Having said this, you should also note that budget hotels will sometimes be without hot water, at intervals without water at all. Be prepared to be without power for short periods of time in a budget hotel, as well.
Regarding
affordable accommodation in the Caribbean, I would say to avoid traveling during high season ( from December to April) if you can as prices are considerably higher during these winter months.
Hope this helps
Vacationbuddy