Hi Tilly
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.
Once you apply for UC and prove your ID, UC will issue a stop notice and ESA will stop, some claimants do receive a 2 week run on with their ESA. You may want to seek help from your local Welfare Rights Specialist; you can find your local Welfare Rights Organisation to help you with advice by clicking on the link below and putting in your post code, it will then come up with a MAP with a list of agencies in your area
advicelocal.uk Gary