Hi Jennifer
Welcome to the forum.
You might want to have a look at the following FAQ which explains where everything is;
www.benefitsandwork.co.uk/guides-for-claimants/faq/forum We cannot answer your question as you have not given enough information such as what type of ESA are you claiming, I know of 3 types, one is means tested the other two are contribution based.
If you are moving to another local authority then depending on what type of ESA you are claiming you may have to claim UC if you need to apply for help with your rent if you live in rented accommodation either in the private or social sector.
As the CAB may have more relevant information, then what they are informing you may be true.
Gary
