Is it possible that stormy weather can be related to sadness? The weather could be related to gloom?
Psychological research has indeed shown that the correlation between bad weather and bad mood is not just a literary clichee but a fact: weather can affect your mood.
While mood cannot affect weather, film makers and writers have used weather to reflect the mood of their characters, as in this scene from The Bridges of Madison County, where a neglected housewife runs away from the possibility of happiness with her weekend lover (played by Clint Eastwood), and as she looks back she sees him standing in the rain and looking after her with what we imagine is longing and despair:
But you must be careful with overdoing this. If the weather always changes in tune with the mood of your protagonist, it will quickly become laughable and disrupt the experience of your readers instead of supporting it.
Restrict your use of the weather to certain key scenes – and sometimes counteract the mood of the protagonist with seemingly "unfitting" weather to emphasize their mood even more:
It's in the language: words like "grey", "dreary" evoke sadness. The sky might be weeping (though that's a little over the top and overused). A lonely seagull might be crying plaintively. Wind might be bighting. Adjectives and adverbs associated with sadness would relate the stormy weather to the emotion you're trying to set for the scene.
Compare that to generous drops falling onto parched earth, washing away some negative whatever. Same rain, different mood. Here, different adjectives set the rain instead as a blessing and a relief.