BT is holding me for ransom.

For last two months, I've been trying unsuccessfully to leave BT as they are getting too expensive. However, I was told that I can't leave with Early Termination Fee. I have given them the reason for my leaving - increase line rental and I was not notified. They told me that they sent me an email to a BT/Yahoo email which I don't use and my account is registered against my gmail. Apparently, I've opted in for email communication. When I asked them to provide evidence, they declined.
Now they are sending me a letter of deadlock and the bottom line is that I have to pay. I would appreciate any assistance with this, e.g. my legal rights and Ofcom rules.

Within how many days of the original announcement of the price rises did you initiate a migration away?

NOTE: They are not allowed to stop you migrating, but you would enter into dispute over any contractual amounts owed.

Have all other communications from BT made it to your gmail account?

If you login to your BT account and go to Settings.

Your BT ID details
These are the details you use to login to your My BT account online. Your BT ID is also the contact email we will use to tell you your bill is ready.

What email address does it show?

In reply to a post by JohnR:

If you login to your BT account and go to Settings.

Your BT ID details
These are the details you use to login to your My BT account online. Your BT ID is also the contact email we will use to tell you your bill is ready.

What email address does it show?

That's in My Profile not Settings.

Just one small thing - while BT changed other prices, they did not change the telephone line rental price.. you needed to quote increased FTTC or other costs.