Hi seasicksteve68
You do not say which ESA you are receiving IR or CB ESA, do you receive PIP?
If you are not going to apply for HB then moving should not force you on to UC.
Council Tax Support Scheme (CTRS) does not fall under UC so claiming CTRS will not force you to apply for UC, you will have to let your new LA that you are only claiming CTRS and not HB element, most LA you can apply on line if you have internet access.
In regards to notifying ESA about a change of address, you need to do this within one month of moving but this can be extended with good cause, don't forget to notify HMRC.
www.gov.uk/tell-hmrc-change-address Gary