  We report $^{77}$Se-nuclear magnetic resonance (NMR) results down to
sufficiently low temperatures under magnetic fields parallel to both the
$ab$-plane and the c-axis in a paramagnetic/superconducting (PM/SC) phase of
K$_x$Fe$_{2-y}$Se$_2$. The observation of anisotropy in the orbital part of the
Knight shift results in the anisotropy of its spin part increasing on
approaching the transition temperature. The anisotropy of the Korringa relation
suggests the presence of the weak spin fluctuations with a finite wave vector
$\bm{q}$, which induce the magnetic fluctuations along the ab-plane at the Se
site. Such fluctuations do not correspond to the stripe $(\pi,0)$ correlation
of the Fe moment observed in many Fe-based superconductors, and are not
contradictory to weak $(\pi,\pi)$ correlations. The nuclear spin-lattice
relaxation rate $1/T_1$ shows a field-independent $T_1T \sim const.$ behavior
at low temperatures for $H \parallel ab$, which is attributed to the nonzero
density of states at the Fermi level and can be explained by the sign-changing
order parameter even for nodeless gaps. The temperature dependence of $1/T_1$
is reproduced well by nodeless models with two isotropic gaps or a single
anisotropic gap. The obtained gap magnitude in the isotropic two-gap model is
comparable to those obtained in the angle-resolved photoemission spectroscopy
experiments.
