  We report a ^{125}Te-NMR study of single-crystalline Pb_{1-x}Tl_{x}Te for x=
0 (undoped non-superconducting "parent"), 0.35 at% (doped but on the borderline
of superconductivity), and 1.0 at% (superconducting, with a critical
temperature Tc ~ 1.0 K). The Knight shift in the normal state is enhanced as x
increases, corresponding to an increase in the average density of states (DOS).
The NMR line-width also increases significantly with increasing x, indicative
of a strong spatial variation in the local DOS surrounding each Tl dopant.
Remarkably, for the superconducting composition, the ^{125}Te nuclear spin
relaxation rate (1/T1T) for Te ions that are close to the Tl dopants is
unexpectedly enhanced in the normal state below a characteristic temperature of
~10 K. This temperature coincides with the temperature below which the normal
state resistivity experiences an upturn. Such a simultaneous upturn in both the
resistivity and (1/T1T) is highly anomalous, and puts Pb_{1-x}Tl_{x}Te in a
distinct new class of doped semiconductors. These observations provide
microscopic evidence for dynamical charge fluctuations in the absence of
paramagnetism, and are consistent with expectations for charge Kondo behavior
associated with the Tl dopant ions. In contrast, such anomalies were not
detected in the non-superconducting samples (x=0 and 0.35 at%), suggesting a
connection between dynamical valence fluctuations and the occurrence of
superconductivity in Pb_{1-x}Tl_{x}Te.
