The relocation money was money that you would've spent if the company hadn't covered it, so yes, that's considered part of your income by the IRS. Nothing you can really do about that.
As for the dual taxation, you're allowed to exclude up to $82k of foreign income each year (and you may be able to either deduct or claim a tax credit on the UK income tax paid) on your federal return. Depending on your state, you may be able to do something similar (because the exclusion is done while computing Adjusted Gross Income, if your state's income tax form starts off with "Adjusted Gross Income from Form 1040", you're golden).
That said, the only issue that could have been avoided before you flew over is the state taxes: it's a good idea to establish residency in a state without an income tax (New Hampshire, Florida, Texas, Tennessee, South Dakota, Wyoming, Nevada, Washington, and Alaska) before coming over, though in your situation that may not have been possible. It's very difficult to establish residency in such a state now that you're in the UK (and doing so may mean having to go through the immigration/work permit process all over again...). The core issue is that while the federal gov't will eventually consider you a non-resident of the USA (though that doesn't get around the fact that income from anywhere in the world earned by US citizens is taxable by the USA... if you're thinking that renouncing your US citizenship is a viable option, note that if you're deemed to be renouncing for tax reasons, the USA will continue to collect taxes from you for 10 years afterwards with the penalty for nonpayment being permanent loss of your right to enter the USA for any reason), states uniformly will consider you to always be a resident of that state unless you establish residency in another state (but not another country!).