ESA withheld after a move

I have been receiving ESA and PIP for a few years now. I was placed in the support group - unsure if that is relevant but mentioning just in case. Recently I moved to a different county - both scottish counties - and I phoned both the ESA and PIP phone lines to inform them. Against advice, I did not follow up these calls with written letters; I did, however, phone a few days later and checked my address status and asked for written confirmation to be sent to my new address which I received from both ESA and PIP.
After this change my first ESA payment came through as normal but a fortnight later the payment did not turn up. I phoned ESA to find out what was going on and I was told that while my address was changed in the system an extra form that was to be sent to my new local council was not filled in and forwarded. Due to this error I could be weeks without a payment.
For context, I am not currently in my own property. I am temporally living with family while I get my new medical team in place (of which I have started making appointments) and get my social worker assigned. Once these things are in place I will apply for my own housing, which to my knowledge will trigger the change on to UC so I am getting prepared for that.
Is this purely the error caused by an uncompleted form on the ESAs behalf or have I made an error myself somewhere? I am unsure how to move forward and I, if it is me at fault, I am worried about repercussions like sanctions.
Any thoughts or advice on this would be appreciated. I don't know that I have explained myself and the situation well so feel free to ask more questions and I'll do my best to answer where I can.

Hi Gordon,
Thank you for the quick reply. I was receiving local LHA and living in a private rental. I contacted the council the day I moved out, however, and I have a letter from them confirming the cancellation of my LHA and the date that it ended.
It is reassuring to know that I am likely not in the wrong here. I am going to contact the ESA again tomorrow for a follow up - to check that form has in fact been sent as I did not receive a follow up call from my local council as promised this afternoon - and if that yields no progression I will go down to the local council service point tomorrow to see if they have any further info surrounding this form as they have a benefits service there.