If the agency is at-risk of losing its funding, then a stand-alone 501(c)(3) is about the only way to go. The agency will need its own status to sustain funding from any source.
If, on the other hand, the agency is supplementing reduced funding, then I would look to the East Tennessee Community Foundation or the Middle Tennessee Community Foundation. These funders have a bit more leeway in deciding what should and should not be restricted funding. By the same token, should the agency completely lose its state department status, one of the community foundations may agree to administer the program under its funding status or may agree to fund the organization's filing process.
I know you know this, but let me reiterate that 501(c)(3) status is not the only issue. Once gone from state, the agency must incorporate in the state as a nonprofit and recieve whatever state deductibility status is available, then the agency may apply for federal status. All of this is predicated upon being to attract a board of directors and incorporators. The issues here of course are time and money. Once closed as a state department, the state will not likely allow the fledgling organization to retain any state funds. Hence the move toward adoption by a community foundation.
(I know you also know this The best place to look for sympathetic foundations would be Tennessee's directory of Private Foundations. (I've forgotten the proper name of the organization.) Somewhere in Tennessee is an association of foundations or grantmakers such as Associated Grantmakers of Massachusetts, Maine Philanthropy Center, etc. The directory will be smaller than what is found on FC and a bit easier to manage (or read through).
Look not only to foundation funders, but also to individual donors. Whether you use a "friends of" org to funnel funding, creating a "friends of" membership program will aid your direct mail/annual appeal campaign. Major donors are more likely to be individuals as foundations.
I gather that the agency does not have much opportunity to "make up" with the state and return to the fold?????