After she gets approved by the Senate she will resign from the Senate and then be sworn in. No conflict there at all.
That's right and things like this have been going on since the beginning.
In the old days, Senators like Webster, Clay, and Calhoun left Congress to enter the cabinet and got their jobs back at the next elections after they left the cabinet. State legislatures chose Senators in those days and so long as legislators kept their seats, they'd be loyal to their state's "big man."