You bring the application form for Rent Supplement to the CWO who covers the address you plan to move to.
The criteria for rent supplement require an applicant to have been renting privately for 6 of the previous 12 months or to have had a housing needs assessment carried out by the relevant local authority, which is the authority which covers the area where you propose to live.
In relation to the housing list, each local authority maintains their own individual housing lists. I'm not sure if you can continue to be on the list in one area if you're living in another - you should check that with your local authority.