From the Florida Statutes:
"617.0203 Incorporation.- (1) Unless a delayed effective date is specified, the corporate existence begins when the articles of incorporation are filed or on a date specified in the articles of incorporation, if such date is within 5 business days prior to the date of filing. (2) The Department of State's filing of the articles of incorporation, and the original recorded charter or certified copy of the charter of a corporation which has not been reincorporated under s.(NNN) NNN-NNNN is conclusive proof that the incorporators satisfied all conditions precedent to incorporation and that the corporation has been incorporated under this act, except in a proceeding by the state to cancel or revoke the incorporation or involuntarily dissolve the corporation."
See also Florida NonProfit Articles of Incorporation and FILING PROCEDURES FOR GEORGIA PROFIT AND NONPROFIT CORPORATIONS
There is no provision that would allow not for profit organization as an LLC in the Florida or Georgia statutes that I see, since it specifically refers to articles of incorporation rather than articles of organization.
A limited liability company is a hybrid business entity having characteristics of both a corporation and a partnership which seems incompatible with the Florida Not-for-profit Corporation Act.
In Florida it is required to change from an LLC to a corporation prior to becoming a not for profit.
Of course, 501(c)(3) refers to the section of the federal income tax code.
Although it is possible for a charity to be a trust, corporation, unincorporated association, or other type of organization it seems that an LLC can only qualify under certain conditions.
"An LLC can be exempt as a disregarded part of an exempt organization that is the sole owner of the LLC."
"An eligible entity (which may include an LLC or a partnership) that claims exemption as a separate entity is treated as an association, rather than as a partnership or disregarded entity."
"The Service will recognize the 501(c)(3) exemption of an LLC that otherwise qualifies for exemption if it satisfies each of the 12 conditions below." (Page 3)
I hope this is helpful even though there is some conflict between the state not for profit and the federal tax exempt provisions regarding an LLC.