We present our NMR studies of double-layer hydrated cobalt oxides NaxCoO2-yH2O (x ~ 0.35, y ~ 1.3) with various Tc = 0 - 4.8 K and magnetic transition temperatures. High-resolution $^{1}$H NMR spectrum served as an evidence for the existence of H$_{3}$O$^{+}$ oxonium ions. $^{23}$Na nuclear spin-lattice relaxation rates served to detect local field fluctuations sensitive to Tc. $^{59}$Co nuclear quadrupole resonance (NQR) spectra served to classify the various Tc samples. From the classification by $^{59}$Co NQR frequency, the double-layer hydrated compounds were found to have two superconducting phases closely located to a magnetic phase. In the normal state and at a magnetic field in the $ab$-plane, two $^{59}$Co NMR signals with different Knight shifts and different $^{59}$Co nuclear spin-lattice relaxation times $^{59}T_{1}$ were observed. The two $^{59}$Co NMR signals suggest magnetic disproportionation of two Co sites or in-plane (XY) anisotropy of a single Co site. Non Korringa behavior and power law behavior in zero-field NQR 1/$^{59}T_{1}$ above and below Tc suggest non-Fermi liquid and unconventional superconductivity.
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