isn't sleepy kind of like tired and half-asleep between awake and sleep? one a statement about your energy level, the other a statement of your operation mode? sleepy: you need/ should sleep. Half-asleep: you are in bed and in the process of falling asleep or in the morning after plenty of sleep but missing a coffee and hitting the snooze button or also just sleepy during the day.
Halbschlaf and schläfrig don't feel like the same thing to me in German.
Not to get stupidly pedantic, but technically the meaning isn't exactly the same. The sentence with an either forces the "or" to be exclusive rather than inclusive--"one or the other, not both" versus "one or the other, or both". Without the "either", though the phrasing would mean it's unlikely ( and the fact that people rarely speak the inclusive or ), it could technically be an inclusive "or".
I know this sentence doesn't use the word "either" but if you would want to put it in the sentence just to make the sentence sound more smooth you would put another "aut" before "semisomnus". Aut...aut would translate to either...or in English so "Marcus est aut semisomnus aut stultus" would translate to "Marcus is either half-asleep or stupid".