Plot hole: For a convicted murderer who violated her parole and assaulted her parole officer while escaping custody, Ashley Judd moves around the country and even boards airplanes with little to no problems.
Suggested correction: She was simply careful. There's constant manhunts for much more serious felons and parolees on the lam who seem capable of moving around without getting caught.
How did she keep the gun if she flew across the country?
She likely put the gun in her bag and then checked it with other passengers' luggage at the airport. As long as she wasn't carrying the gun on her, it would go through.