As a dual member of a state society you have all the rights of a regular member of the State Society and can hold offices in either or both of the State Societies at the same time, but you cannot be a voting representative of both Societies at the same Regional or National meeting.
You may join a Chapter in the dual state if you wish or if it is required by that State Society.
National dues are collected through your primary state and not through your dual state. You will pay for only state dues (and perhaps Chapter dues) in the dual state.
Get a copy of the form at the link below. You may print the resulting filled-in form on 8.5 by 11 paper. This does NOT have to be special watermarked archival paper; any regular paper will do.
You should contact the State Secretary of the State Society in which you wish to become a Dual Member. He can inform you of the correct fees and procedures, or put you in touch with the correct person who handles Dual Membership Applications for that Society.