Switched from Income to Contributions

I received an updated council tax bill where I was told my council tax support was reduced due to the DWP changing me from income related ESA to contribution related ESA. Has anyone else experienced this? I can't keep having benefits taken off of me.

The amount of ESA money I receive has not altered at all, but, following a change of address, the council have increased how much I pay in council tax, even though the property move was no change in terms of banding. It is this increase, which I am challenging, that has opened up the need to confirm which payment of ESA I am in receipt of, IR or contributions.
I am also in the support group, so thought this dictated I am in the IR type.