I've been told I have to claim UC. I'm on ESA but moving to a different council. Is this correct?

As described by you it is not correct. Moving to a different address, even in a different local authority, does not require you to end your ESA and switch to UC.

However I assume that what you have not told us is that as well as ESA you are claiming Housing Benefit. If you change local authority you cannot take your Housing Benefit with you. If you want help with rent you have to claim UC. The act of claiming UC ends your income based ESA. If any of your ESA is contribution based it will continue (but will reduce the amount of UC you receive).

it is correct that income based ESA will continue for two weeks after you claim UC. it is also correct that your ESA status will transfer across to UC.

If you do need help with rent then there is no need to apply for UC.