I beg to differ, I was only able to watch the first video you posted and I'm not seeing anything to cause major concern. She is not coughing, nor appears to be stressed out, in fact by looking at her, she seems to be quite calm and in overall good shape.
Your female budgie does not look to be in distress to me, she is doing some soft regurgitating motions in order to bring up seeds and then she ends up by swallowing them back. This is normal behaviour in budgies and other pet birds, they would sometimes tease and trick their mates into pretending to feed them, this is part of their playful games, they even use this during the courtship rituals. What your budgie girl did is a perfect example of this.
She also tries to feed her toy bell a bit by the end of the video.