If a us citizen marries a uk citizen and moves to the uk do they still have to file taxes witb BOTH the usa and the uk if they get a job in the uk?

Yes.

A U.S. citizen must file a annual U.S. income tax return each year as long as they retain U.S. citizenship. It is likely they will pay no taxes due to a combination of the Foreign Tax Credit (Form 1116) and the Foreign Earned Income Exclusion (Form 2555), but the return STILL has to be filed.

Further, if they have foreign financial accounts (bank accounts, brokerage accounts, insrance plans, annuities, etc.) that, in total, exceed $10,000, they must also submit the FBAR (for Foreign Bank Account Report).