You need to understand how DNA works. Women pass mtDNA down from mother to daughter. Is your mother the daughter of Elvira Riggs? If so, then your mother's mtDNA is the same and your mtDNA is the same as Elvira's.
Is your maiden name Hawkins and you are wanting to trace it? If so, since that is your father's name then you need a male to test for the Y-DNA.
The newer autosomal DNA tests can help you trace your family back approximately five generations on BOTH sides of your family. If your parents are alive, or better yet, grandparents, then it would be better to test them.
Deciding which service to use for DNA testing is a personal choice. You should read what they each say and make your decision. I personally like FTDNA, but that's just me. Lots of people use them, but lots use ancestry also.