Elinor is about to write a letter to Edward when he walks in.

Elinor would not ignore propriety, nor would Col. Brandon, who asked her to tell Edward of it the living, due to his own delicacy, suggest that she do so on his behalf.

Obviously, a one-off letter, to answer necessary business, WAS allowed between a man and woman. (In Austenuations I could also point out that Mr. Darcy wrote an explanatory letter to Elizabeth; Mrs. Weston corresponds with Frank Churchill; etc.) Jane Austen herself wrote a letter to the non-publisher of Susan (which later became Northanger Abbey) signed Mrs. Ashton Dennis.