I used to work for LTSB and it's ridiculous that you're getting the run around. You should be able to merely change your address for your new address. As an existing customer they should require nothing more than
1. Your signature if you hold a signature on file on a Change of Address form.
2. You should be able to do it over the phone if you hold a phone bank security number.
Don't talk to them about it, just say "I want to change my address" and give them your US address.
If you're having issues, request that they add a correspondence address to your account whilst keeping your UK address (don't worry no post will ever go there) as your home address. That will ensure all post goes to your US account.
I think you may have trouble with your cards though, LTSB have recently switched to using a secure courier system for a lot of their deliveries, which doesn't cover the US.
They may elect to send it to a branch in the UK for you to pick up, they SHOULDN'T do this, but that doesn't mean they wont.
But seriously, LTSB is one of the easiest banks for changing addresses, they have no problem with overseas addresses at all, and even have a US/UK dollar account.