Do I need to complete beneficiary forms to designate my legally-married same-sex spouse as my beneficiary?

No.  Beneficiary forms are not required if the order of precedence meets your needs:  

Order of Precedence:  If there is no beneficiary designated, to your widow or widower; if none, to your child or children, with the share of any deceased child distributed among descendants of that child (a court will usually have to appoint a guardian to receive payment for a minor child); if none of the above, to your parents in equal shares or the entire amount to your surviving parent; if none of the above, to the executor or administrator of your estate;  if none of the above, to your other next of kin as determined under the laws of the state where you lived.

If you wish to designate a different order, you must complete Designation of Beneficiary forms.  If you do complete Beneficiary forms, it is your responsibility to keep them up-to-date to reflect your current wishes.