We report systematic measurements using the Co-59 nuclear quadrupole resonance (NQR) technique on the cobalt oxide superconductors NaxCoO2.1.3H(2)O over a wide Na content range x = 0.25-0.34. We find that T, increases with decreasing x but reaches a plateau for x <= 0.28. In the sample with x similar to 0.26, the spin-lattice relaxation rate 1/T-1 shows a T-3 variation below T-c and down h, to T similar to 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 similar to 2 K even though the T-c (similar to 4.7 K) is similar, which suggests an unusual evolution of the SC state. In the normal state, the spin correlations at a finite wavevector 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.
