  We report systematic measurements using the ^{59}Co nuclear quadrupole
resonance(NQR) technique on the cobalt oxide superconductors Na_{x}CoO_{2}\cdot
1.3H_{2}O over a wide Na content range x=0.25\sim 0.34. We find that T_c
increases with decreasing x but reaches to a plateau for x \leq0.28. In the
sample with x \sim 0.26, the spin-lattice relaxation rate 1/T_1 shows a T^3
variation below T_c and down to T\sim T_c/6, which unambiguously indicates the
presence of line nodes in the superconducting (SC) gap function. However, for
larger or smaller x, 1/T_1 deviates from the T^3 variation below T\sim 2 K even
though the T_c (\sim 4.7 K) is similar, which suggests an unusual evolution of
the SC state. In the normal state, the spin correlations at a finite wave
vector become stronger upon decreasing x, and the density of states at the
Fermi level increases with decreasing x, which can be understood in terms of a
single-orbital picture suggested on the basis of LDA calculation.
