I've thought about it. Two things come to mind. One possibility is that Ted left behind a tear-stained copy (ghost copy?) of the first letter Doris wrote to Bax, the "Dear John" letter, which had been written on a sheet of Ted's old stationery. (That is nothing, compared to giving your dead husband's wedding band to another man.)
A better possibility is that it was a blank sheet of his stationery. That is, she was telling the truth (maybe) about Ted wanting her to hook up with Bax; his tears on it were his testimony. She added her tears to his and sent the second, big-mistake letter, possibly even on the same sheet of paper left by Ted, the dried tears bearing silent witness to her sincerity. Bax had done something similar earlier, getting blood stains from the beating by Ieuan on a letter to George.
True, Ted didn't leave a note, exactly, but tears on a sheet of stationery would say something.
-Roy