This one’s all about the commas.
Here’s the corrected version:
The moon cast an eerie glow, and snowflakes fluttered lazily around the trees. She sat on the forest floor, playing a haunting tune on her ebony flute.
The first sentence is compound: two short sentences joined by a conjunction (and). That means you need a comma before the conjunction. (No, there is not some sort of exception to commas in compound sentences, such as ‘only use a comma with but, not with and.’) The comma in the second sentence is for clarity. If the sentence had said, She sat on the forest floor next to a wolf, looking hungry, for example, you’d really want that comma so you’d know that she, not the wolf, was the one hoping for dinner.