Do you have to get a pardon of the felony to be able to vote or run for office?
Run for office at the national level? No, there is nothing in the Constitution that would prevent it. Now Tennessee law may prevent her from voting for herself as a convicted felon but she's free to run for the Senate.
Her actions since her conviction have shown she's turned her life around. But count of Alexander to make hay of her past if she appears to be any sort of reasonable threat to him in the primary. And, of course, the Democrats will try to hammer her on it should she win the nomination.