Define “the south” for your purpose of what states or regions you want to avoid.
Big merit NMF/NMSF schools and their specialties mentions Fordham University, Washington State University, University of Minnesota Twin Cities, University of Southern California, University of Maine that are not in what most people would define as “the south”.
There is also mention of University of Tulsa and University of Missouri, which may or may not be “the south”. Of course, your home state of Virginia and its universities like Virginia Commonwealth University mentioned there is also often considered to be in “the south”.
Also, the net price after the NM based scholarships does vary, so it may be that some are too expensive even with those scholarships, which may be automatic with NM status or competitive beyond that. Also, not all of those college with NM based scholarships assure admission to those with NM status.