Thanks for the feedback! @ChrisNordgren
I hear you regarding them looking a bit stale. My intention was to create something stylized without a ton of crazy movement in hopes that it would come together more fully within a particle system while still maintaining a somewhat stylized look. I'll see how far I can push this in future iterations.
Regarding the wiggle, it was created based off of a trick fellow VFX artist Dare Matheson showed me. It is a UV offset that is distorting the U by using vertical sine waves that pan upwards. It looks like the following in it's most basic form and can be used for UV offset or vertex offset:
Now you can beef that portion up by adding additional sine waves into the mix and providing controls to alter the height and width of the texture as well. You can lerp two sine waves of different speeds and length by a third larger sine wave to get some nice variety or provide controls to blow the fire in a certain direction to simulate wind using the same principles (for this you could use a float, multiply that by a similar mask that masks the base, and add this to the sine wave right before you add it into the R Mask. A negative number for the float will push it one way and a positive number will push it the other way).
As far as the look of the fire, I use two textures to form the basic shape. One that serves as a sort of blob fire with no holes in it and another that acts as the mask and defines the detailed shape of the fire (just as Chris guessed). The masked texture scrolls upwards and both have the same UV wiggle distortion added to them. For further variation you can add a little more random (non wavy) distortion to the masked fire to help change it's shape a bit.
Let me know if that doesn't answer your question or if I can better explain anything!
I'm continually playing around with this method and will post any updates if I am able to make any improvements to it!