Fighting a cold and coughing several times during his remarks, Mr. Biden sought to tie Ms. Loeffler and Mr. Perdue to President Trump’s failed efforts to overturn the elections by raising false allegations of voting fraud in Georgia and other swing states. Mr. Biden mocked the two senators for supporting a lawsuit filed by the Texas attorney general that sought to overturn the results of the presidential elections in four states, including Georgia. In an email to his supporters as he flew to Atlanta from Washington, Mr. Biden made a raw, political appeal for campaign contributions to help the Democratic Party win. 