I delayed answering a bit to do some research myself, just to make sure. There’s definitely a connection to Elune, but her phases have no visible effect on worgen in the game or other sources.
In a classic werewolf story, the moon and the werewolf curse are closely connected. The werewolf bite may not take effect until the next full moon, some werewolves only change on the full moon, and - in some - it is the full moon and the bloom of a Wolfsbane flower that creates a werewolf.
But as for the worgen, the worgen bite takes effect in a matter of a few hours (Mardigan, Dark Riders) to three days (Halford, CotW), the change occurs “when driven to anger”, and the first worgen were elves given the curse by Elune.
The one thing that I can find, regarding Goldrinn and the phases of the moon, is the Kaldorei legend that Goldrinn would spiral into furious bloodlust on the full moon. When Malfurion Stormrage and Tyrande were scolding Ralaar and Arvel on the use of Pack Form, they told the druids that Goldrinn hates Elune. Or, at least hates the way she looks on him with disappointment. They claimed that he was an angry beast to begin with, but when her light shone full and bright, it was as if she glared down upon him for his noble potential lost under his needless bloodshed.
Whether this story of Tyrande’s is true or not, I don’t know. Her bias is clear, no one had heard this story before, and it’s never backed up. But that is one solid example of “full moon = beast mode”. If you wanted a character that did the same, you could give them a similar personal relationship with Elune, or a strong connection to Goldrinn. Such a connection that does cause extra rage, as seen in Varian Wrynn in Wolfheart, though his anger was not dependent on the phases of the moon.