Must I buy the Udhiyah out of my own money—that saved for my childbirth expenses—on behalf of my husband, who has limited income, and am I sinful if I do not?

The Udhiyah (sacrificial offering) is a confirmed Sunnah, not obligatory, according to the preponderant opinion of the majority of scholars. It is required from one who is able to afford it without it causing a significant financial burden on their wealth. If you are in need of the money saved this year, or if your husband's income is low, then it is not required of you both.