Putative quantum spin liquid triangular materials, NaYbO₂ (powder) and NaYbS₂ (single-crystal) are investigated by bulk magnetization and specific-heat measurements, as well as by ²³Na NMR and Yb ESR techniques as local probes. No signatures of long-range magnetic order are found down to 0.3 K for both materials, evidencing a highly frustrated spin-liquid-like ground state in zero field. Under external magnetic fields ranging 3-7 T along the triangular lattice plane, signatures of magnetic order are observed in thermodynamic measurements, suggesting a possibility of an anisotropic field-induced quantum phase transition. The ²³Na NMR relaxation rates reveal the absence of magnetic order and persistent anisotropic magnetic fluctuations down to 0.3 K at very low fields, and confirm the bulk magnetic order above 3 T, field parallel to the triangular lattice plane, by the critical slowing down behavior of the nuclear relaxation, T₁. The H-T phase diagram for the magnetic order is obtained and discussed along with the existing theoretical concepts for layered spin-1/2 triangular-lattice antiferromagnets.