GENERAL TIPS:
Adult cartoons should touch adult topics.
Adult cartoons are not about edgy/sex jokes.
Adult cartoon requires you to take your audience seriously.
Every part of the adult cartoon needs to work in harmony with eachother to produce a meaningful result.
If you want to write a good adult cartoon you need to have a:
-INTERESTING WORLD SETTING.
Most people watch cartoons to escape reality and prefer to see something special instead of nigh 1:1 copy of our world, take advantage of that by spicing up your world to make it feel more unique.
First of all you need to choose in which era your plot is set in and go off that, then you can move on to more important things.
You can replace humans with Anthropomorpic or Zoomorphic characters.
You can add magic, or mutants, or something else, there are tons of possibilities and they are limited only by your imaginations and laziness.
-CHARACTERS THAT AREN'T CARDBOARD CUTOUTS.
They are the the back bone of your plot and are the ones who move it forward, your job as a writter is to make them feel alive.
Like in the real world, people are different, they have different interests, they strive for different things, they have different habits, they have secrets, they can have flaws, they have different traits that make them who they are, they can change, same aplies to your characters.
It doesn't matter if it's a villan or a main/side character, people love seeing a thoughtout characters that persue their own goals, think for themselves and develop over the course of the story, that's called a character arc.
Remember your characters are not perfect, flawed characters are more interesting to watch than bland and perfect ones.
Be causeous when following a common character tropes it's a double edge sword that either make your character feel awesome and flashed out, or make them unbearable and cliché.
-CAPTIVATING PLOT.
Second most important thing is the plot, what exactly drives your characters to do things they do.
Most of the time plot starts with a simple goal that grows larger over the course of the story, there could be many reasons for the plot to start, but YOU MUST go past the simple goal of doing or finding (something) and switch to more and more complex things. It can be for example character development and relationships, those two affect the plot drastically and can dictate the decisions characters make, which means more deviation from the main goal and thus more captivating plot.
-ALL ACTIONS HAVE CONSEQUENCES, BOTH GOOD AND BAD.
Speaks for itself.
Your characters actions should not exist in a bubble, it's pretty simular to Newton's third law in a sense that if your characters do something bad it can result in something even worse both for them and a world around them.
Same thing vice versa, characters can do something good which will reflect both on them and the world around them.
Some actions can mix both, for example character can perform something horrible which will reflect badly on him but will make world around him better.
-BALANCE BETWEEN HUMOR AND SERIOUSNESS.
It's REALLY important to choose which one has more weight on the scales, but usually your plot and targeted audience dictates that.
You need those two things to make your cartoon bearable, people love good character development, yes, but that doesn't mean they want to watch only that, same with jokes, people love jokes, but they also want to see something more serious and meaningful. A good show usually mixes both.
For the love of god don't think that because you have an "adult cartoon" it means you can joke about sex, almost if not all people hate that and will actively try to avoid it.
You should base your jokes around context, or make it so your character references something funny, important to note you should focus more on old times humor, simply because humor nowdays just sucks, and making references to whatever kids like nowdays is generally a bad idea.
I am touching a tip of the barrel here, there is just too many things to keep in mind, but you will develop a writters sense on your own and will be able to see which things fit and how to do them