I work for one of the big six (not SSE or Scotish Power)
I fail to see why you can't just pay it online or over the automated system, then do the switch to whatever provider you want (through Uswitch or using referral link directly through the other providers website). Unless you havn't bothering giving your start readings so everything is estimated, which is entirely on you if thats the case.
Also you cant/wont be cut of unless you have PAYG/Prepay.
You wont get anywhere sending letters/emails to "headoffice" whatever you think that means, you really think they are going to bother? they are not customer facing.
Also dont pay by budget direct debit if you can get away with it, pay by direct debit whole amount monthly so you are not giving your energy company more money that what is necessary to pay your bills.
Its a new move I literally have the meter readings and that's it, only today out of luck I could get through to provide them and let them know the property isn't empty (had been for a while).
I guess people so used to the big shixts poor service that's its standard practice now. Not quite sure how people are spinning the fact that it's somehow acceptable to be almost completely uncontactable when providing essential Services.