Here’s the 2nd earliest Snow White story document I have, and I just noticed that these early documents are from this month, October, only 85-years earlier.
By the time of this document, it’s amazing how much effort has already gone into Snow White’s story construction, and you’re really starting to see the character personality development.
A trick they used was to not simply start-out creating a new personality from scratch, but to mimic and build upon an already well known live-action personality. Before the process was over, though, most of these characters did evolve into their own unique personalities, removed from the original live-action reference.
It’s a fascinating document (click on image below to open):

This was fascinating to read.
Interesting that the Queen makes two attempts to poison Snow White in this one. I can see why that got changed – it’s really longwinded and makes Snow White falling for two tricks makes her way too gullible.
One bit that made me laugh was the Queen transforming into the hag – with warts popping out of her face and then hairs popping out the warts. While it would have been funny, I’m glad that the gags were ultimately left out of this scene.
The prince has A LOT more to do at this stage. Some of his actions sound pretty corny, dated and melodramatic, but it’s cool there were attempts to make him much more of a character.
As it went through the process, the story was chiseled down to a single thread that wove itself throughout the entire movie. Each action moved naturally from one to the next, propelled by reason — they didn’t just make up a convenience to move on; they questioned every action’s motivation and relevance. I don’t think there’s any other Disney movie that went through this much story development — almost to perfection.
I just watched Frozen again, and although I thought its music and visuals were superb, a lot of the story points were forced and sloppy. None of this sloppiness in Snow White.
Also, in Snow White, as you alluded, the gags ended up mostly being with the Dwarfs; everything else was played relatively straight. A conscious developmental refinement.