Below is a direct copy from American Ancestors as to the father of Eunice Dodge - note her name is spelled Unice.
DODGE: Unice, d. Ebenezer, bp. Sept. 28, 1712. C.R.1. Town Page Detail: Beverly Birth Beverly, MA
Additionally in the genealogy book I mentioned regarding the parents of Ebenezer Dodge, you will find, on page 7 not only her birth, but the fact that she married Benjamin Fellows of Ipswich, and they had three children.
I hope this has answered your question as to the parents of Eunice Dodge bef 28 Sep 1712 - 19 Jul 1747