I can't locate the passage I remember about Varina Davis - possibly it's only in the unedited edition and not the searchable one online, or possibly my memory is in error.
I did find this passage which mentions a wet nurse in Mary Chesnut. She seems to think it is a usual thing.
Thursday. - I find I have not spoken of the box-car which held the Preston party that day on their way to York from Richmond. In the party were Mr. and Mrs. Lawson Clay, General and Mrs. Preston and their three daughters, Captain Rodgers, and Mr. Portman, whose father is an English earl, and connected financially and happily with Portman Square. In my American ignorance I may not state Mr. Portman's case plainly. Mr. Portman is, of course, a younger son. Then there was Cellie and her baby and wet-nurse, with no end of servants, male and female. In this ark they slept, ate, and drank, such being the fortune of war. We were there but a short time, but Mr. Portman, during that brief visit of ours, was said to have eaten three luncheons, and the number of his drinks, toddies, so called, were counted, too. Mr. Portman's contribution to the larder had been three small pigs. They were, however, run over by the train, and made sausage meat of unduly and before their time.