Hi, welcome to Just Answer. I will help you with your question.
I've had this problem with Three personally, so I might be able to assist on a very practical level also.
Three can transfer a contract to a pay as you go number, which then allows them to transfer it back to a contract. They don't like transferring contract numbers to a new contract number for some reason. However, the pay as you go option is a very easy way of addressing this.
As for the cancellation period, this should be possible via contacting three and not having to visit their shop. As I understand it, there is a 14 day guarantee period, where you can cancel with them. This is a matter of contract - it's not within the general law, so what they promised you about cancellation when you signed up is important.
When they talk about cooling off periods not applying when you buy from shops, this generally means that you have no rights under Distance Selling Regulations, which is correct. However, Three offer something above this (or certainly, they used to) and this is to cancel within a period, even if you buy from a shop.
So, I expect notification to them by telephone to cancel is sufficient for your purposes. Sadly, the truth is, you often have people selling these phones and working on call centres that don't understand the law or even their own policies.
If you have tried to exercise a promised right of cancellation and they refuse, then you can sue them for your money back. You can do this via the small claims procedure: http://www.scotcourts.gov.uk/taking-action/small-claims
Can you see the chat?
I'll switch to question and answer mode, which you might see more easily.
thanks for your input but I have just talked to three again and the received the same trans continental gobbledgook.I shall just pay up a three be damned.