No they don’t die, but some mourning doves do show signs of acknowledging the loss and will fly to the same spot repeatedly were they lost their mate and even tend to and try to take care of the deceased bird if the body is still present.
Here is a thought. Mourning Doves mate for life. Maybe she lost her mate. I don’t know what they do when they lose a mate but she could just be grieving or “lost”.
Just be prepared to have people on here give you a hard time with keeping her and not taking her to a rehabilitation center. It’s illegal to keep wild native animals. With that being said, if I were in your shoes, I’d keep her too. Do lots of research on their diets as she will wanna eat what...