the local authority have a duty to help, either with council property or refer you to a housing association, or they put you in temporary accomodation,you will be homeless soon so i would make the local housing office, my first port of call
also shelter may help, also contact your local housing association.
it does appear if on benefits your children under ten can share even if opposite sex and under one is nil points, whether the same applies to private i do not know. look on line for private landlords and property agents in wirral.
tell all the relevant people you will be homeless from a certain date. hope this helps.