No. Sarah is no longer The Duchess of York. Like the ex wife of any other peer she uses her Christan name and then the female version of her former husbands title so she is called Sarah, Duchess of York without the "The". If Andrew were ever to marry again his new wife would be HRH The Duchess of York while his ex wife would still be Sarah, Duchess of York.
What Sarah would call herself after remarriage would be anyones guess. She would not be the first ex wife of a peer who continued to use her former husbands title. Raine Spencer comes to mind. She married and divorced a French count and after the divorce she resumed being known as Raine, Countess Spencer. Presumably there is more cache in using a well known British title than a French one. Since Sarah makes a living off of being Andrew's ex wife it would not surprise me if she continued to be known as Sarah, Duchess of York for commercial reasons even if she married a Mr Smith.