I was recently interviewed along with some of my colleagues in Classical Singer magazine about the issue of degrees, and some of the pros and cons of advanced degrees like the DMA, etc. There is a wide range of viewpoints on degrees for singers. On the one hand, some feel that a doctoral degree may indicate that you either lack talent as a singer, or are not serious about your pursuit of a serious singing career. Some people feel you shouldn't even list them on your resume because of that. I don't have any prejudice against a particular degree from any particular institution, and I am proud of the degrees I earned. At the end of the day, my belief is that it is your audition, and not what is listed (or not listed) on a piece of paper, that will get you hired.
Back when I was doing university teaching, there was a companion belief that you shouldn't let anyone know you had a teaching job, because that would weaken the impression you made in an audition. For the record, I never hid my teaching position either.
I guess where I land on all of this, is that everyone has a story to tell of why they are where they are, and how they got there. Your life story and musical journey may be such that a DMA program is perfect for you. On the other hand, you may have a chemistry degree, and you've become a wonderful singer purely through private study. The point is, only you know what is the right path for you to take. An no matter what path that is, nothing can replace having to produce the goods when you get to the audition.