First, if a Judgment for Dissolution of Marriage was entered, the State in which it was entered generally retains jurisdiction with respect to child custody matters. You should seek a modification of that portion of the Judgment which granted shared/joint custody. The current court order remains in effect until a subsequent court order is granted.
Contact the Court where your Judgment was entered and they can direct you in the proper procedure if you cannot afford to retain an attorney.
First, I would caution to whether or not it will be wise to sign over your legal custody rights to the minor child. Just because Father would be deploying overseas does not require him to sign over these rights. While sole legal custody will provide Mother with the flexibility to make decisions while Father is overseas without consulting with him, it also provides Mother with the flexibility to utilize that decision making ability in a way that could negatively affect Father should she chose.
With sole legal custody, Mother could enroll or terminate the minor child's enrollment in any private or public school, could start the child on any number of medical, therapeutic, and/or even psychiatric treatments without consulting with Father, [even by email (which may be available to you at different times during your deployment in various countries)]. Sole legal custody also may permit Mother to relocate her residence, and change the area of the child's primary home without Father's consent. Also, depending on whether or not this is a temporary order, or a "Montenegro" (trial) order will also weigh heavily into your decision. The burden required to change the legal custody back when you return home safely may be more difficult than you think in the event that it is a "Montenegro" or trial order.
In the event that the parties are amicable, and you decide to proceed with the "signing over" of your legal custody rights, consult with a local family law attorney, and have them prepare a stipulation and order. This will need to be signed by both parties, and filed with the court. This is fastest means to which you can accomplish your results.