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Geometrically, $|z+5-3i|$ is the distance between the complex numbers $z$ and $-5+3i$ in the complex plane, and $|z-7+2i|$ is the distance between $z$ and $7-2i.$



By the Triangle Inequality, the sum of the distances is minimized when $z$ lies on the line segment connecting the two complex numbers $-5+3i$ and $7-2i,$ in which case the sum of the distances is simply $|(5-3i)-(-7+2i)|=|12-5i|=\boxed{13}.$