Shouldn't be difficult to figure out a time -she and her sister moved in with Grandmother when she was 8 years old.
She started having noriety by appearing on various Tv shows at ages 22-23 The MC and IC tells the story as to when she was born.

Here's some stuff about Carol that may help in fine tuning her birth time:

Both of her parents, particularly her father, suffered from alcoholism, and at a young age she was left with her grandmother

When Burnett was in the fourth grade she created for a short time, an imaginary twin sister named Karen, with Shirley Temple-like dimples. Motivated to further the pretense Burnett recalled fondly that she "fooled the other boarders in the rooming house where we lived by frantically switching clothes and dashing in and out of the house by the fire escape and the front door. Then I became exhausted and Karen mysteriously vanished."

Burnett's mother disapproved of Carol's acting desires: "She wanted me to be a writer. She said you can always write, no matter what you look like. When I was growing up she told me to be a little lady, and a couple of times I got a whack for crossing my eyes or making funny faces. Of course, she never, I never dreamed I would ever perform."

She married Don Saroyan on December 15, 1955; the couple divorced in 1962.
Burnett's first true taste of success came with her appearance on Broadway in the 1959 musical Once Upon a Mattress. In the same year, she became a regular player on The Garry Moore Show. She won an Emmy in 1962 for her "Outstanding Performance in a Variety or Musical Program or Series" on the show.

On May 4, 1963, Burnett married TV producer Joe Hamilton, a divorced father of eight, with whom she had three daughters: actress and writer Carrie Hamilton, Jody Hamilton and singer Erin Hamilton. The marriage ended in divorce in 1984, and Joe Hamilton later died of cancer.

Carol Burnett Show debuted in 1967, and didn't cease production until 1978.

Burnett drew attention in 1981, when she sued the National Enquirer for libel after the tabloid newspaper described her alleged public drunkenness, purportedly with Henry Kissinger. Burnett was particularly sensitive to the accusations because of her parents' own alcoholism.

In 2001, she married Brian Miller (principal drummer in the Hollywood Bowl Orchestra), who is twenty-five years her junior.

Personal tragedy struck Burnett in January 2002 when her daughter Carrie Hamilton died of lung and brain cancer at the age of 38.

She was known for ending each show by tugging her ear, which was a message to the grandmother who had raised her to let her know that she was doing well and that she loved her.

Carol has always been 'reachable', never above her fans, never refusing to give an autograph.