Adoptees dont have a "long" birth certificate.They have the short one that you mention and another called an extract from the adopted persons register. Both will be in her adopted name.
She needs to get both her extract and the original birth certifiacet with her birth mothers name on it (unless she was a founding). It rarely has the birth father's name.
The extract will give the date of her adoption.If she was legally adopted. i dont mean that her adoption was illegal, just that many people who belive they were adopted were simply brought up by relatives and there was no legal process or involvement of an adoption agency. Sometimes "adoptions" were arranged informally by doctors or ministers of religion.
I'm just wondering if that might be the case withyour mother. Partly because it was common in teh 20s and 30s and partly becase she only has her short birth certifiacte. Though it might be simply that her adoptive parenst might not have wished her to know that she was adopted
For example, a baby born to a teenage unmarried mother coudl be "adopted" by grandparenst or an aunt and uncle.. I knwo of a case where a child was informally adopted by another family, but the "adoptive" fathers name was put on the birth certificate along with the birth mothers. then the baby was simply brought up by her legal father and his wife
There is counseling availabel for adoptess who are searching and your mother may wish to take advantage of this, as it can be a difficult process. I think thsi is especailly true for people of your mothers age, who were brought up wth different attitudes to these matters