When writing drama, aim for this:
And avoid this:
Ironically but explicably, drama can be a massive source of laughs. This happens when drama is not done well—when what the characters take seriously is farcical to the reader.
How can you avoid this, though?
First, to write any kind of drama, you yourself are going to have to FEEL something, too. The best way to get into a dramatic moment in your story is to write for an extended period of time, and don't pause to edit. Write exactly what your heart tells you, even if you think it's "too much" or "melodramatic," because you're only going to be able to get into an emotional zone about this scene one time probably: when you first write it. Let your emotions write the scene.
If you're anything like me, writing good drama is going to drain you. If you're a good writer, you'll also be bothered that—because you're writing the scene so quickly—you're missing a lot of details and the result will appear rushed and unfilled.
Which takes us to the second part. Reread your drama. On your next pass-throughs, fill out the drama with details to pace it. Pacing is mainly what you want to fix here. Also tone down anything that seems "too much" (silly, melodramatic) but without compromising the emotional tone. You're trying to find a balance between melodrama and no drama.
Drama is time sensitive, and again you're really only going to get one or two more shots to edit it well. This is because if you read drama too many times, you're not going to feel the drama anymore. It'll get stale to your taste. Editing drama too many times will lead it to be understated. In your eyes, the drama will become easy to communicate very subtly. You'll be unable to divorce yourself from the more bombastic drafts, and you'll lose perspective on what a first-time reader will think. So don't over-edit drama. Edit 2-3 times, or at least quit dialing it down after that point. And as we've discussed before, give at least your final edit some space, 3+ months without reading your story.
When you write drama and edit drama, be sure you have the time and energy to do it. Avoid cliches and things you "see in the movies" if you can, although of course tropes can be strong if used tastefully. Let literature be your guide and the stories that touch your heart. What did the writers of your favorite stories do to achieve what they did? Take notes, and emulate their methodology.