At the outset, we are not critical about your score but by the fact that you are embarrassed by it.
Pls understand just like good days, bad days-there are good scores/score that can be improved.

So, we request you to focus on the following questions:

(1)Do you have a chance to appear in other rounds later for GMAT?
(2)Is there a possibility for you to appear for GRE?
(3)Assessment of your loopholes-Analyze the points that can be given a correction. For example are you fully prepared with the basics or you still keep going back to observe the concept videos/chapters before you solve a question and/or when you find yourself stuck at a question?
(4)How good you are at anxiety and pressure management? Are you able to solve every question under 2-2.5 min with precision?
(5)Have you taken enough mock tests? How have you been performing in them? Have you analyzed your errors?
(6)Have you done the OG?Are you confident that if any question is asked from OG ,you can answer it correctly under 2-2.5 min?
(7)Is your revision up to date?

Once you have assessed the above mentioned, you can have a broad idea of how to restart again. You may think of taking assistance of a mentor. Touching 600 is not impossible in the time limit, when you are aware of where you can fix the issue(if any).
This said, we urge you once again to understand that of all the things in the world, scores are the last things to feel embarrassed of.
If you create a "hawk eye" perspective of the concerns at hand, we are sure you will come out with flying colors and nothing could make us more happy!

 We wish you all the best.

You have provided an elaborated response and we are confident that you must have also realized some key areas to work on(on a broad note). We are happy to provide you the following inputs!

(1)On one hand we shall suggest you to leave no stone unturned and think through GRE as well and on the other hand, if you make up your mind to appear for GMAT only then we suggest you to pull up the gears and improve on all the aspects and full proof a target score.

(2)You have mentioned it yourself "I might not be doing correctly: I don’t really know how to assess my weaknesses."
You have also mentioned that you have taken the GMAT twice and you are yet to secure your target score. We suggest you take help of a mentor-
anyone you feel and know are reliable and can guide you by analyzing your scores professionally. You have increased chances of knowing exactly
where you need the guidance and hand holding.

(3)You also mentioned "Precision-wise, I feel I might not be the most organized person when it comes to solving a problem." You may miss a few of the higher/advanced level questions but if the precision factor is not taken care of well in the easy/medium levels it shall affect your score by a big margin. Hence accuracy and accuracy under pressure are key notes of a perfect delivery.

(4)You take the daily quizzes and it is appreciable. However mocks give your mental senses a simulated testing atmosphere. When you do well, you feel confident and get the boost in the next mock and if you falter somewhere, you can analyze where you need to find the ground. Either ways, taking mocks is essential to full proof your prep. It keeps you on your toes and makes you feel on the line always. Hence we suggest you with all our experience to not negotiate with taking mocks on a fixed schedule and most importantly analyzing them.

(5)You mentioned about the OG. You also mentioned some questions are easier for me and some aren’t.! And we totally understand this!
That is the scheme of the OG. Having said that, it is those questions that are your speed breakers where you can go through and train your brain
to think on. The more you think to figure out solution, the more your mind stays tuned to find solutions and look for them at the
right data points.. So we suggest, practice-think on the speed breakers-Repeat on a loop. Exercise the brain cells there!  

(6) You have mentioned I keep re-writing formulas and tricks I have learned for better retaining. That's amazing!!
It is essential. We can suggest you here to organize this review/revision day and make sure that a weeks/half a week's work is revised on
this day! Make it a habit to NEVER skip the Revision Day.

Now that you know , you are not alone and you certainly have no reason to be embarrased of..Gear up and REBOOT for a better, fun and target
 oriented journey.