I spent time digging around in newspaper archives, because I was curious and hadn't seen anyone else answer. Unfortunately, a lot of the newspapers from that period are not available outside of a paywall. So I’ll quote some passages to give you a sense of what they look like.

A little background: news took a while to get to the south. While Willie died on Feb 20, the earliest reference I found, in a Confederate paper, was February 28, and most didn’t cover the story until early March (3rd, 4th), and the latest was March 8th. Most northern papers carried the story almost immediately (most commonly February 21st and 22nd).

There are essentially three stories that were carried by the Confederate papers. The vast majority of the stories, and the ones most frequently reprinted, are factual and respectful.

Story #1, which appears in the Daily Selma, Charleston Daily Courier, Richmond Enquirer, Richmond Examiner, and Richmond Whig is short and succinct: “Willie Lincoln, second Son of President Lincoln, died on the evening of the 20th. In consequence of this event the public buildings and the private residences of the Heads of Departments in Washington were not illuminated on the evening of the 22nd instant, as previously arranged. The youngest son was also dangerously ill, and it was feared could not recover.”

Story #2, which appears in several North Carolina papers, Charleston Mercury, and the Richmond Dispatch is a little more colorful: “Amid the general joy excited by the success of the Union cause, a black shadow has fallen upon the Presidential mansion, and all who were personally acquainted with the family of the President share in the deep tried occasioned by the death of little Willie Lincoln. He was a boy of such promise that all who became acquainted with him had predicted for him a career of no ordinary character. Young as he was, he impressed every, one who came in contact with him as a youth who was destined to become a man of rank." Some of these accounts also included information about embalming and the doctors involved with some additional context.

Story #3, which appears unique to the Southern Confederacy (which was out of Atlanta) reports the information factually but is unable to resist a dig at the Union’s expense (but not about the President’s son): "William Lincoln, second son of the President of the United States, died on the 20th of February, and another son was seriously ill. This affliction befell when preparations were rife for an unusual grand celebration of the 22nd of February, the birthday of George Washington, the first and greatest of American 'rebels,' whose name and example are now prostituted to the purpose of tyranny infinitely worse than any which Washington and his compatriots resisted."

Definitely an area of study that would be interesting to explore more, but hopefully this offers some useful perspective to begin with.