UC and moving house

Can u pls clarify an issue for me pls. I am moving house shortly, within the same borough i now live in. I asked someone at the housing benefit office if i would have to be moved onto UC, she said no, but admitted she wasn't too sure and gave me the UC helpline which is impossible to get thru to. So i turned to B and W for advice and u said no i wouldn't. I have recently been reading other members posts who share the same concern as me, and they were told by u that they would be moved if they were making a new claim for any of the benefits that fall under the UC umbrella. I'm so confused that now i don't know if i want to take the chance of moving. I am currently in receipt of CB ESA (IR top up) support group is that CB ESA or IR ESA and i get the full SDP included also enhanced daily living PIP. Many thanks to u all.