  We report a $^{11}$B NMR line shape and spin-lattice relaxation rate
($1/(T_1T)$) study of pure and lightly carbon doped MgB$_{2-x}$C$_{x}$ for
$x=0$, 0.02, and 0.04, in the vortex state and in magnetic field of 23.5 kOe.
We show that while pure MgB$_2$ exhibits the magnetic field distribution from
superposition of the normal and the Abrikosov state, slight replacement of
boron with carbon unveils the magnetic field distribution of the pure Abrikosov
state. This indicates a considerable increase of $H_{c2}^c$ with carbon doping
with respect to pure MgB$_2$. The spin-lattice relaxation rate $1/(T_1T)$
demonstrates clearly the presence of a coherence peak right below $T_c$ in pure
MgB$_2$, followed by a typical BCS decrease on cooling. However, at
temperatures lower than $\approx 10$K strong deviation from the BCS behavior is
observed, probably from residual contribution of the vortex dynamics. In the
carbon doped systems both the coherence peak and the BCS temperature dependence
of $1/(T_1T)$ weaken, an effect attributed to the gradual shrinking of the
$\sigma $ hole cylinders of the Fermi surface with electron doping.
