I think it can be done, but I don’t know how the Argent Dawn community looks at it these days. I play my priestess with a hunter class. Mind, I specifically underpowered my character; She can still call upon the goddess, but her blessings come in the form of convincing animals to fight on the sentinels’ side.
I don’t know exactly how to do that with a druid played as a priestess. But a backstory is easy to imagine. Maybe your priestess was in Val’sharah to defend the Temple of Elune and therefore witnessed the terrible fall of Ysera. This event shaped your character to want to know more about Elune’s connection with nature, so you’ve been learning a limited amount of nature magic, without intending to become a druid. A way to honor Ysera. Maybe centuries in the future, she will become a full-fledged druid, but that is certainly not today.
I don’t find it unbelievable that a priestess of the moon might have become a druid of the moon. Female druids would have had to be pulled from somewhere and from novice priestesses does make a lot of sense. In the lore, characters switch to a new class all the time, though they tend not to mix and match their powers that often. A mage can become a warlock, but once they are a warlock, they tend to focus themselves on being a warlock.
Think about it like this; Druids were asleep for millenia and yet the sentinels had to interact almost exclusively with wisps, treants, ancients and other such creatures to create their mortal dwellings. That means that they always had to have used nature magic to some level, but they wouldn’t consider themselves a druid for it.
Druids are people who wander the world, healing it from it’s grievous injuries, whether they do so in the Emerald Dream or in the physical world. Their scope is broad to the point of being alien. They may have a fondness for night elf society, but their reach is much farther than Ashenvale. When the druids went to sleep, they did so because they were done fixing Ashenvale and needed to use all of their focused efforts to fix the rest of the world after that. A priestess simply wouldn’t do that; she has a parish (though a priestess wouldn’t call it that, aside) to consider and a civilization to build.
The dichotomy that I make is; “priestess = social engineers”, “druids = environmental engineers”. They might use the same tools, but their objectives are just very different and complimentary.
TLDR: You can be a priestess of Elune who is dipping her toe into nature magic, without intending to go all-in (at least, that’s what she’s telling herself). Night elves canonically can use nature magic regardless of what class they are. It’s their version of technology. Blood elves use arcane magic regardless of class and night elves are pretty much the same. Kind of. It’s complicated.
Unfortunately, public perception can make it hard to portray well-thought out character concepts. People have certain ideas of what elves should be and look like. It’s up to you and the community whether you want to continue putting up with that. I find it incredibly frustrating that the community most fanatical about portrayals of the night elves has the least tolerance for verisimilitude.