Council housing is in short supply everywhere but once you get on the list they can give you address's of housing associations and put you forward for housing.
Do be advised that in many areas of England, council housing and housing association housing are now one and the same thing. In an increasing number of instances, a council will transfer its remaining housing stock to a housing association in return for the Treasury writing off the council's debt w/regard to that stock.
It's worth looking into, but it's definitely not the sure fix it once was.
Children under 10 count as 1/2 a person w/regards to overcrowding, and I'm afraid you'll find many, many families living in technically overcrowded housing who haven't got much of a chance of being rehoused.
Sorry to sound so grim, but we went through this a few years ago ourselves.
Much of council's quality stock was sold under Right to Buy. The councils were not allowed to plow the procedes from such sales into replinishing stock, so affordable homes to rent were lost.