In this work, we present an adiabatic and nonadiabatic study for all the (1)Sigma(+) states dissociating below the ionic limits [i.e., Na (3s, 31), 4s, 3d, 4p, 5s, 4d, 4f), K (4s, 4p, 5s, 3d, 5p, 4d, 6s, 40 and Rb (5s, 5p, 4d, 6s, 6p, 5d, 7s, 4f) + H (1s)]. The ab initio calculations rely on pseudopotential, operatorial core valence correlation and full valence CI (configuration-interaction) approaches, combined to an efficient diabatization procedure, For the low-lying states, our vibrational level spacings and spectroscopic constants are in good agreement with the available experimental data. Nonadiabatic potentials and dipole moments are analyzed, revealing the strong imprint of the ionic state in the (1)Sigma(+) adiabatic states, while improving the results. This paper can be used as theoretical support for experimental work.