For dominant hand, the only word I can think of which is synonymous in context is preferred.
Right-handedness: The word adroit is defined to mean dexterous, deft or skillful. As the OP has noted, dexter is Latin for "on the right". Digging a little deeper, the etymology of adroit reads thus:
1650s, "dexterous," originally "rightly," from Fr. adroit, from phrase à droit "according to right," from O.Fr. à "to" (see ad-) + droit "right," from L.L. directum "right, justice," accusative of L. directus "straight".
Furthermore, the French word for right-handed is droitier and maladroit ("inept; clumsy; awkward; not adroit"), while rarely used, can also mean left-handed. The more mainstream term for left-handedness in French is not all that much better, as it is gauche ("Awkward or lacking in social graces; bumbling.").