DWP won't change me over to ESA assessement rate

I have an appeal in progress and asked DWP to move me over to ESA assessment rate from the JSA I was claiming.
I last signed on the 23rd January and went in to my advisor on the 31st Jan because my sick note was running out and he told me that there was a note on the system to state that they had received my appeal and to inform the me, 'the client' that he has the right to now move over to ESA and asked me if that was what I wanted to do. I said that I did, and he told me that I no longer had to sign on and for now, that was all I needed to do.

Anyway, I did not sign the following week, as I was told I didn't need to and lo and behold, my payment did not go through. I rang to chase up the payment and was told that the money was not released because I did not sign on, despite being told that I didn't need to.
This has been going on for about 6 weeks now, and I keep being told to come in on what would normally be my singing day and sign on anyway in order to still get my money.
I have asked them so many times to move me over to ESA because of the stress of all of this and each fortnight, it is still the same.
I have had to make numerous phone calls and they just keep telling me that I cannot claim ESA without closing down my JSA claim. They want me to completely shut down my JSA and then start a new claim for ESA but something about this just doesn't seem right.

I was always told that I would be moved over to ESA automatically so long as I inform them that this was what I wanted, which I did. I refuse to shut down my JSA because I do not want to have my housing benefit suspended for another 9 weeks like before, while they sort this out!

I don't understand what they are saying to me, because how can I make a new claim for ESA when I am not entitled to it, which is why I am appealing.

Can someone please tell me if this seems right, because they must be dealing with this sort of thing often and the advisors seem to be really misinformed about what the procedure is. One keeps saying, no, we don't shut down the JSA claim, we just transfer them over to ESA assessment rate. Others are saying, we can't get you on ESA until you shut down your JSA first and are pressuring me to either shut down the JSA or start doing proper intensive job search again since I am still claiming JSA.

Please help!! I am so stressed and do not want to mess up my money. My rent is nearly due and I am being told to do something which could affect my housing benefit.
When they disallowed my ESA when I failed my medical, I had to claim JSA, they stopped my rent for 9 weeks because they failed to confirm that I was changing over to another benefit, not having my benefits cancelled and so I was without money for months. The fact that I had already informed the council of this impending change, meant nothing because the DWP just said, 'no longer entitled to ESA', instead of 'changing from one benefit to another'.
Thank you for your time.

Thank you Gordon. Last and final question......So I get them to close down my JSA and then claim ESA. Will it be a new claim that I have to make, or will they reinstate my ESA at assessment rate without having to make a new claim?
I am not understanding how I can claim ESA when I have been disallowed it, hence the reason I am appealing the decision?

Hi Gordon, Can I ask a hypothetical question just to clarify the changes to new claim ESA payments after 3 April 2017:

If this happened to Jane a month from now, and her claim was reinstated at Assessment rate, and then eventually her appeal was upheld ... would she be backpaid the WRAG component, and continue to receive the WRAG component?

Or is there the chance that having closed her JSA claim and opened a ESA claim, that claim would be considered a new ESA claim which was not eligible for the £29.05 WRAG component?

Mathew