The humor in general is really fucking out of place. Soren fights off assassins, fails to protect his king, and what are he and his sister concerned about? Coffee.
It's not necessarily that the quips are the problems, it's the timing of them and some weird idea that's cropped up nowadays that you need a joke every five seconds no matter what the characters should actually be thinking and feeling in that moment. A good example of this for me is in Thor Ragnarok, an otherwise perfect blend of comedy and grimdark violence, when Asgard is destroyed and there's this brief moment of silence until... there's a fucking joke.
When the King died, the mood in the capital should have been absolutely abysmal. Everyone should have thought they had failed and maybe Claudia could have tried to cheer her brother up with the coffee, but it should only have half worked with a depressed "Thanks..." instead of a "OH WOWSERS, THIS STUFF IS GREAT! THEY SHOULD SELL IT ON EVERY STREET CORNER!" The King is fucking dead. Act like it.
And what's even more frustrating about all this is that the show does have a lot of good writing in it, especially in the overall plot structure. Where as ATLA fucked around going from one town to the next for no goddamn reason, DP was relatively deliberate about where it set its filler and both the snow lodge and the mountain town felt like they advanced the story far more than they just lengthened it. And the subplot with the arm binding was pretty good while it lasted, and I'm sure it was supposed to last even long but some exec or weak coward among the writing team had to have insisted it be resolved by season 1.
Where the Dragon Prince succeeds is in its overall vision for the story, and where it fails is in the moment to moment stuff. It's a frustrating contradiction, but it's definitely something that can be fixed in the second season.