Hi Zinny,
Housing service charges can be paid with ESA or with UC.
If you move but stay in the same Local Authority area then you are not obliged to claim UC. But if you move to a new Local Authority then you have to claim UC for housing costs.
We cannot go into granular detail regarding service charges as the topic does not fall within the remit of the BenefitsandWork forum
David
Nothing on this board constitutes legal advice - always consult a professional about specific problems