If the Elves are more godlike than mortals, well, any given mythological god is prone to very mortal foibles. So must the Elves be. They bleed, they feel anger, love, hatred, envy. In these ways they are much like mortal Men and Dwarves. Ah, but there are differences, and important ones! Largely, these differences are a matter of
perspective. Elves hunger as do humans. They craft things as do humans. But their perspective is fundamentally different, being closer to the gods by their very nature, immortal and keenly touched by Middle-earth's unique mysticism.
We all know that Middle-earth isn't as flashy as most high fantasy. Mythic resonance takes on a real life of its own, has a power and an agency every bit as magical as a fireball or a flaming sword. Deeds and intent have real effects beyond the immediate; it's not unlike the Force in Star Wars in this way. There is magic in raising a garden, say, for the simple pleasure of tending the growing earth. If you do evil things, and have an evil heart, those forces might literally corrupt you until outwardly you resemble that corruption. The Rings were made with a desire to preserve or, in Sauron's case, to dominate others. Magic is pervasive but often subtle, infusing the words, actions, and feelings of the entities of Middle-earth.
Elves are more keenly aware of this than others. This is how you separate them when they have obviously mortal qualities. You describe them as something beyond mortal but not quite god. The Elves woke up the trees, Treebeard says, but how did they do that? They weren't necessarily through big, flashy spells. It was more likely through a simple desire to communicate. We get a few examples in the books that really highlight this:
Are these magic cloaks?' asked Pippin, looking at them with wonder.
'I do not know what you mean by that,' answered the leader of the Elves. 'They are fair garments, and the web is good, for it was made in this land.... Leaf and branch, water and stone: they have the hue and beauty of all these things under the twilight of Lórien that we love.... Yet they are garments, not armour, and they will not turn shaft or blade. But they should serve you well: they are light to wear, and warm enough or cool enough at need. And you will find them a great aid in keeping out of the sight of unfriendly eyes, whether you walk among the stones or the trees. You are indeed high in the favour of the Lady! For she herself and her maidens wove this stuff; and never before have we clad strangers in the garb of our own people.'
Elves made magic cloaks indeed, but they didn't do it through vast arcane energies. Rather, they simply poured some of themselves, the things they loved, into their craft. The result was a magical item, enchanted by any stretch of the imagination, but it wasn't really a deliberate thing.
Quickly Sam unslung his pack and rummaged in it. There indeed at the bottom was a coil of the silken-grey rope made by the folk of Lórien. He cast an end to his master. The darkness seemed to lift from Frodo's eyes, or else his sight was returning. He could see the grey line as it came dangling down, and he thought it had a faint silver sheen.
...
'It's good to be able to see again,' said Frodo, breathing deep.... 'I could see nothing, nothing at all, until the grey rope came down. It seemed to shimmer somehow.'
'It does look sort of silver in the dark,' said Sam.
Here we can see another subtle bit of magic. The mere presence of something from the good folk of Lórien, imbued with their subtle mystical qualities, dispels the darkness clouding Frodo's eyes. And of course the rope comes undone in accordance with Sam's desire. It was as strong a knot as anyone ever tied, but because he wanted the rope to come undone when it did so that he could reclaim it, the rope did just that.
How do you maintain the mystery? For one, you will most likely have to trust your players to do it. Give them the benefit of the doubt regarding playing Elves. The other is by things like this, demonstrating a perspective fundamentally different from those of mortals. The Elves make magic simply by
being, enchanting cloaks and ropes by doing nothing more than putting a piece of themselves in their work. A mortal Man might do the same thing, but not end up with an enchanted item, or indeed even conceive that it is possible. But the Elf keenly aware of the hidden forces of the world, sensitive to the flows of time and fate and meaning, to her it's not even "magic." It's just a consequence of their act.
Mythic resonance is key.
Meaning itself has real power in Middle-earth. Elves embody this more than the other peoples. They can speak with stones and wake up trees simply through the desire to communicate. The very act of putting the thought of things they hold dear into what they create enchants their creation. They utter curses that will shake the foundations of the earth. They fashion starlight into gems of immeasurable beauty. Their love of nature is such that a bit of soil from Lórien can help a humble gardener from the Shire, whose love of the green growing earth must surely rival their own, transform his beloved homeland. They even had golden-haired children born after that point, something that had been extremely rare up until that point.
To play an Elf and maintain that sense of wonder, particularly High Elves, I suspect that the in-character perspective must take these things into account. To the Elf, song and action, feeling and will, these things are all the same. Expression is the direct conduit from the Elven soul to Middle-earth. The Elf character doesn't cast a spell that is under her control but channeling a force separate from herself. She is Lúthien, taking mastery of Sauron's fortress through sheer will. She is channeling
herself into the effect.
Maintaining this alien perspective while contrasting it with the similarities is paramount to portraying Elves as PCs. The Elf PC dreams but does not (always) sleep. Hungers but does not starve. Feels pain, but does not age, or grow sick. The Elf sees and feels things that the others can't, because the Elf character is simply more in tune with the forces that shape Middle-earth. Stress that at every good opportunity and I think you'll be golden.