Hi Sarah
As long as you do not need to claim help with housing costs then at this moment in time you will not have to apply for UC, you just need to inform DWP and your local authority of a change of address.
Most claimants who receive ESA SDP will be worse off applying for UC.
This information does come with a caveat, there are a number of areas across the UK which are now affected by UC, so I would check with your local Welfare Rights Organisation if your area is affected;
advicelocal.uk Gary
