In seeing that conceptualizing or thinking doesn't alter my state, I don't feel the need to for it and I don't expect it to make me feel better or expect a solution.
Strangely, I had a few moments of insight where thoughts were just recognized without giving them meaning. Without going into the content, I noticed that every thought plays out automatically, there's no need to do anything with them. I became confident that the thought will play out the moment I first saw it. The moments of noticing the beginnings of thoughts became just a series of uniform meaningless arisings. I was quite surprised at how fast these can occur. I started to notice moments (of clarity?) between arisings and then I found that I don't know how to tell the difference between the moment of clarity and the arising, because the recognition itself is just another meaningless arising. Then I somehow got overly excited about it and it was over
Anyway, I wonder if that was the glimpse of nonduality.
That's good! However, In seeing that conceptualizing or thinking doesn't alter your state, and that you don't feel a need for it...
Inquire into the nature of the one who believes these things.
Thoughts do play out automatically, everything is spontaneously arisen in the frame of considering 'things to arise'. The thought is liberated as soon as it appears. Seeing the thoughts as a series of uniform meaningless arisings is also useful, it's 'seeing' them in their 'suchness' as 'sounds' or 'noise', and eventually they won't be 'seen' as sound or noise and won't even arise or fall but will be timeless expressions of the true nature. But don't let that 'lead' you to seek that, just continue with your investigation. Expect nothing. Rest in those moments of 'clarity' (or whatever name you want to give them) getting excited or thinking they represent 'progress' will be counterproductive, if excitement arises immediately perceive that as a 'meaningless arising' as well, and let it self-liberate, continue to rest unmoved. The 'moments of clarity' will strengthen and 'become longer' if that makes sense, and you're right there ultimately is no difference between the clarity and the arising. But don't believe what i'm saying... just look for yourself... and don't let it 'validate' your conclusions... just continue to rest in that clarity.
Not a glimpse of nonduality, because it seems from how you described it that you still feel "you" are witnessing this 'happening'. What can help with this is to start to notice the habit of assuming there is 'time'... or a 'series of arisings' as you put it. Question this notion of time. Question the notion of 'space' between 'you' and an 'arising'. See that this notion of you(subject) witnessing arisings(objects) depends on these notions of 'time and space' and inquire into the nature of 'time and space'. Or locate 'that' which is untouched by 'time and space' and rest there, all of this falls in line closely to what you've already been saying so don't look too hard.
Apologize if it sounds like i'm preaching or teaching; don't believe a word i say... just look! Investigate empirically.
(All of this is my own opinion, i attach no authority to it.)