The glass transition and relaxation dynamics in the binary mixtures of a Debye liquid, N-ethylacetamide, with water, monoalcohol, and amine are studied by calorimetric and dielectric measurements in the highly viscous regimes near the glass transition. Calorimetric measurements show the glass transition temperature in the N-ethylacetamide-water mixtures is remarkably enhanced as water is added as high as 70 mol. % before crystallization is detected. A similar increase is also observed in the N-ethylacetamide-rich mixtures with the non-Debye 1,2-propanediamine. However, the dielectric measurements show that the main relaxation in the N-ethylacetamide-water mixtures with water fraction up to 60 mol. % reproduces the dynamic characters of the mixtures constituted by two Debye liquids, N-ethylacetamide and 2-ethyl-1-butanol. The comparison of the calorimetric and dielectric features for the three mixing systems suggests that the Debye relaxation persists in the N-ethylacetamide-water mixtures within the experimentally studied compositions.