Is your mother-in-law's sister deceased? If yes, are there any records pertaining to her that might contain any clues? (A will, an obit, a funeral record, a cemetery record?)
It does seem odd that she would have had to go through an official adoption process, when her name was on the birth certificate.
Wouldn't the mother's name on the birth certificate have to be amended if it was discovered or admitted legally that it had been falsified? (If so, both the mother's given name and the surname would have had to have been changed, because your mother-in-law's legal surname was her married name.)
Did anyone else in the family have the same given name(s) as the woman who raised (and presumably adopted) your husband's cousin?
Did your mother-in-law have any other sisters who were old enough to have had a baby at that time? Or any cousins?
You don't necessarily have to have older relatives to ask: sometimes people tell their children the family secrets. I've heard a few stories about someone not knowing s/he was adopted but all their cousins (or some of their cousins) knew, because their mothers told them at some point.
Did your mother-in-law have a will? Aside from her funeral wishes (I don't understand what is meant by having four readings instead of three), did she make any provisions for your husband's cousin?
Good luck!