There are 1 bedroom rentals in New Jersey, but not sure if they come with 2 kids...
Seriously, there are plenty of 1 bedrooms rentals available in NJ in many different price ranges, and I am assuming you mean you want to put yourself and both kids in that 1 bedroom?
Contact a local Real Estate agent in the areas you want to look for an apartment, and he/she can help you find something in your price range. Many landlords pay commission so you won't have to worry about that fee.
You may not need to get 1 bedroom and squish 3 people into it, and honestly, some municipalities have rules on occupancy depending on the number of bedrooms/square footage of the apartments etc, so you may NOT be able to get 1 bedroom for three people anyway..