We had to do the same for our daughter when she went to study overseas a couple of years ago. We actually lost the original, and got a copy from the Texas Dept of Health. That was NOT good enough for her to get a passport. It had to be a certified copy from the state.
The wife had to do the same for her passport when she went on a cruise last year. She had to get hers from New York.
I realized when my oldest needed one for driver’s ed that we didn’t have original birth certificates for the two children born in Texas. Fortunately, the instructor accepted the hospital certificate as proof of age, and we got the birth certificate in time for her to take it to the DMV for her permit. I guess I just forgot to order them back in 1991 and 1994!