Defense lawyers in the Georgia election interference case against former President Donald J. Trump seek to disqualify the prosecutors. They aim to prove that the romantic relationship between the lead prosecutor and her hired lawyer started earlier than claimed, suggesting a conflict of interest. New witness testimony from a deputy district attorney and text exchanges with a former law partner of the hired lawyer have raised questions about the timeline of the relationship. The judge is expected to rule on the disqualification within two weeks. If successful, the case against Mr. Trump could face significant disruption.