“The laundry list of salacious & flat-out false reporting about Mrs. Trump by tabloid publications & TV shows has seeped into ‘main stream media’ reporting,” Ms. Grisham wrote on Twitter on Sunday. “She is focused on her family & role as FLOTUS — not the unrealistic scenarios being peddled daily by the fake news.”

A year into her husband’s presidency and her own tenure as first lady, Mrs. Trump finds herself in an unusual position — and perhaps at a disadvantage. There are few things Mrs. Trump can share about herself without it being dissected — often negatively. When she revealed the tidbit that her favorite TV show was “How to Get Away With Murder,” for example, the show’s star, Viola Davis, did not dispute a joke that the first lady was “a captive in her own home.”

The polarizing nature of her husband’s presidency has also isolated Mrs. Trump from her predecessors. She is not part of a small group of first ladies, including Michelle Obama and Laura Bush, who have developed a bond based on knowing what it is like to be constantly scrutinized, with their popularity linked to their husbands’.

“First ladies from Jackie Kennedy to Hillary Clinton to Laura Bush have stood by their husbands at the lowest points in their presidency,” Kate Andersen Brower, an author of “First Women: The Grace and Power of America’s Modern First Ladies,” said in an interview. “We’re seeing a different example with Melania of a woman who has maybe had too much.”

Mrs. Trump’s trip on Thursday to the Holocaust Memorial Museum was, coincidentally or not, in the same direction as Joint Base Andrews, where she was headed to leave for a whirlwind trip to Palm Beach, Fla. While Mr. Trump was still at the World Economic Forum in Switzerland, Mrs. Trump went to Mar-a-Lago, the president’s Florida estate. On Friday, onlookers at the resort were directed by the Secret Service to move their vehicles to ensure a wide berth for Mrs. Trump, who needed a secure path from her residence to the spa.

Attendees at a safari-themed fund-raiser held at the resort on Friday had hoped for an impromptu visit by the first lady, but were told Mrs. Trump had left just before the event began. In her stead, guests had to be satisfied with a giant portrait of the first lady, which failed to quickly sell at auction. (A portrait of Mr. Trump sold for $17,500.)

“She had to leave just as we were starting,” said Terry Bomar, the event’s organizer. “The Secret Service made everybody stand inside as she was coming out.”