Using the Five-hundred-meter Aperture Spherical radio Telescope (FAST), we search for H I narrow-line self-absorption (HINSA) features in twelve Planck Galactic cold clumps (PGCCs), the starless core L1521B, and four star forming sources. Eight of the 12 PGCCs have detected emission of J = 2–1 of cyanoacetylene (HC3N). With an improved HINSA extraction method more robust for weaker and blended features with high velocity resolution, the detection rates of HINSA in PGCCs are high, at 92% overall (11/12) and 87% (7/8) among sources with HC3N J = 2–1 emissions. Combining the data of molecular spectra and Planck continuum maps, we studied the morphologies, dynamics, abundances and excitations of H I, CO and HC3N in PGCCs. The spatial distribution of HINSA is similar to that of CO, implying that HINSA features are confined to regions within and around CO emission kernels. HINSA tends to be not detected in regions associated with warm dust and background ionizing radiation, as well as regions associated with stellar objects. The L-band continnum and average background H I emission may be non-ignorable for the excitation of HINSA. The abundances of cold H I in PGCCs are approximately 3 × 10−4, and vary within a factor of ~3. The non-thermal velocity dispersions traced by C18O J = 1–0 and HINSA are consistent with each other (0.1–0.4 km s−1), larger than the typical value of HC3N (~0.1 km s−1). Carbon chain molecule (CCM) abundant PGCCs provide a good sample to study HINSA.