ESA Payment Suddenly Stopped - No Notification.

Hi there. I am new to the forum, but have joined up in the past, in order to get help with my PIP claim.
In short, I would like to kindly ask for some assistance regarding an issue I am having with my ESA. I have been on this benefit since February this year, and although I attended one medical with a nurse (that was very distressing), I was told I needed to be assessed by a doctor instead.
Since then, I have been awaiting a new medical assessment appointment, so not yet put into a 'category'.
Since February time, I have been supplying medical certificates, and had had regular fortnightly payments, until last week when the payment suddenly stopped.
I couldn't understand why, as I had not received ANY notification that there was any problem, nor any text either.
I phoned them, which was difficult as I am hard of hearing, and after ten minutes a lady explained that my previous sick note had run out, and as it is the sick note that releases the payment, it had been stopped. I told her I had not got any written or text reminder, as I had one time prior, and she told me that they had sent out two letters, neither of which I received, and advised me to get a backdated fit note put into local job enter and put through as an urgent payment (taking no more then 24hours.
I did as advised, my husband put the sick note into the job center, and 24 hours later, the promised payment was still not in my account.
Fearing that I was running out of time before the bank holiday weekend, and getting extremely low on food, I phoned them again. This time, I was holding on for over one hour, when eventually I spoke to a man, who among other things, told me they had two 'systems', and on one my new address was noted, but on the other my previous address was still there. As I had recently moved house, I called them in February to update them of my new address at that time, and was told it had been put on the system.
Needless to say the two letters they sent me, didn't reach me because they sent them to my old address, which is totally they're error, The man said he would now update that error, and he would endeavor to get another person to call me urgently to sort out my payment.
I never got a call, and my payment which was due on 25th of April, still isn't in my account.
I have multiple chronic health conditions, so cannot work and rely heavily on this payment to buy food, and have a defendant child also.
Please can you advise wether they have the legal right to just stop my payment, given they sent the reminders to the wrong address, and thus I actually got no written notification that I indeed required a new sick certificate?
Regrettably, during my house move, I seemed to lose the photocopy of my previous sick note, so was unsure of when it ran out, but thought not to worry, as they would text and write to me when a new one was needed.
I think it is disgusting to do this to anyone, but I am also quite fragile, suffering from major depressive illness, and do not need this added stress.
Should I write a letter of complaint to them about all this? Is it legal for them to just stop paying without any contact being made?
Hope someone can help me and thanks.
Ingrid.