Well, It seems that in a couple of episodes Prince Adam was really insufferable. Like in the classic (they say so) Into the Abyss (minute 4:13). I don't know if he did this because he was really committed in maintaining the facade or because he changed his personality when he was Adam.
I don't see that at all. As he says to Cringer a few moments later, he's just tired of all the fighting as He-Man
and all the studying he has to do as Adam, and he needs a break from the stress. And worse, Teela and his parents don't know he carries that extra burden of fighting all the time to save the world, so they mistake his well-earned need for recovery time for simple laziness. I can't blame him for insisting on some me time once in a while.
I think there's some pretty nice character writing there in the parts I watched. It does a good job of making everyone's motives understandable and sympathetic.
One can make the case the people shouldn't really expect from Clark Kent (a journalist and newscaster) to be a dashing daredevil hero and from Don Diego De La Vega (a rich landowner) to be a fearless swordman because, well, it isn't exactly in their job description.
But Adam is the heir to an absolute monarchy, with specific duties and responsibilities.
It's not like lazy, layabout princes are unprecedented in history. The problem with being a hereditary monarch is that you don't
have to be worthy of your status; it's just handed to you. Plenty of royals and nobles have been useless, self-indulgent idiots. It's inevitable. That's why dynasties fall, as a rule. The founders of a dynasty may earn their power through cunning strategy and military might and the brilliance to build an empire, but each successive generation that's born into that power and luxury has less need to be strong and smart and capable, so eventually you end up with a succession of pampered, imbecilic rulers who only care about themselves instead of the people, and the kingdom deservedly falls. It's why monarchy is a terrible system of government. (Personally I prefer to assume that Eternia is a constitutional monarchy. I don't see Marlena going along with it otherwise.)
And in theory he could even one day eventually lead lead his soldiers into battle (as a fake King Randor did by leading the kingdom's army in an attack on Grayskull Castle). So, when Teela presents her grievances, it isn't like when Lois whines about Clark ("Oh Clark, why you can't be more like Superman?!?"), but more like objective concerns about the future of the kingdom.
Of course Teela's objections are fair, from her perspective. That doesn't mean Adam has a different personality from He-Man, though.
