Eric Dane's Exes Should Grieve, But Family Says They Want the Facts Straight

Rebecca Gayheart and her daughters understand Eric Dane's ex-girlfriends are grieving his loss -- and they don't want to diminish anyone's feelings -- but they want the facts straight about his final days.

TMZ previously reported ... Eric was single for a while after he became ill ... and on his deathbed, it was just Rebecca and his two teen girls.

A source close to the family tells TMZ ... they "feel a deep responsibility to ensure that his story and memory remain accurate" about who was "present until the very last chapter of his life."

We're told the family -- especially his daughters -- wanted that crystal clear because they deserve the space to mourn without being "confronted by narratives that simply don't align with the truth they lived." Basically, they appreciate the condolences ... but not perceived ambiguity or embellishment.

Dane's exes, Janell Shirtcliff and Priya Jain, have both shared their feelings about his death -- Janell a few days after his death -- and Priya recently told TMZ she is still processing the loss. She specifically wished his family well. Worth noting, she also donated $10K to his GoFundMe created for the girls.