Flying with an newborn

We are supposed to go see my husbands grandfather 1 month after our baby is due. We wouldn’t be making the trip if it was a dire situation but since he is sick we want 1 more good visit with him, and for him to meet his great granddaughter. Has anyone flown with an infant 3-4 weeks? I’ve check some of the air lines and most only require the infant be 2 weeks old. My family (which doesn’t fly ever) is telling me I can’t take her on an airplane that young and I should just stay here with her. Obviously it’s our choice but now I’m nervous the doctor will tell us not to go. The only other option is to drive from GA to CA and that seems just a bad.