Is the area around my mouth puffy due to perioral mounds or nasolabial folds? (photos)

This is not the best picture because it doesn't show the whole face.  I see what you are saying.  I think that putting filler in the NLF will exacerbate the problem.  Lifting the area by placement of filler in the cheeks is the approach I would probably do.

This photo is really hard to tell because the lighting is also bad which makes the mouth look more pronounced than it probably is. So, based solely on this photo, I would suggest fillers into the upper cheek region which will help lift that area that is puffy. Adding filler into the mouth and NLF area most likely will make it look even more pronounced than it already appears. If you can provide more photos, we may be able to give better answers. "This answer has been solicited without seeing this patient and
cannot be held as true medical advice, but only opinion. Seek in-person
treatment with a trained medical professional for appropriate care."