Once there is the beginning of a resting in oneness, I have noticed a few different dangers that begin to happen to the experiencer of the non-dual state - or at least to this particular experiencer.

The largest is the fusion and confusion between the the enlightened state, experienced as it is, and the perceptions from this place aligning with various incorrect concepts of the separated self. 

I don't believe this gets enough attention from the various teachers out there, at least as I see the issue.  Since I view this as incredibly important, I hope I can do justice, in words, to communicate this.

a. Surrendering the sense of personal responsibility in inappropriate ways.
b. The separated self "owning" or attempting to capture the sense of this pulse, as itself.
c. A "too soon" confidence in the midst of experience, that causes the regeneration of the personal sense of self as the prime mover of reality.
d. Awakening reinforcing the separated self's tendency to isolate. 
e. Awakening as a panacea for living, without attending to the business of life, and the needs and flaws of the separate self, with the appropriate tools learned as "skillful means". 
f. Confusing the release of awakening, with the specialness of YOUR life.  (While in actual fact, awakening is ordinary, and "just this", the initial recognition is felt as a tremendous event of the timeless, invading the time-bound.)  Zen has this right with the "chop wood, carry water" comment.  While an awakening experience MIGHT ease some psychological distortions or resistance, as an example, one still might need therapeutic help.  Awakening is NOT a panacea for psychological issues.
g. In counterpoint to "e", getting too caught up in the business of life and not attending to the infancy of awakening, so that THIS becomes the ground of your life, rather than the details of the dream you are immersed in.  (This may be a restatement of "c" above, actually, in that this regenerates the prime mover of life as the separate self.)
h.  Confusing awakening as the endpoint of spiritual transformation, and thus ignoring the "Divine Heart", for lack of a better word.  The "transforming of all the 4 bodies", in service and love, to become a "light unto the world" - especially in the ever-quickening hurly-burly self-absorption of the world - is also essential.

I'll ruminate more about this a bit later.