AWA is not part of the score, and I strongly recommend reading this page: https://www.mba.com/mba/thegmat/teststructureandoverview

You should regard the score for exactly what it is, an indicator of where you are right now. If your goal is 680, clearly you are not going to get there without studying. So I am not sure what you are asking in your last question.

I took the second GMAT PREP practice test last night, and I scored a 590 (13 incorrect on quant, 17 incorrect on verbal). I didn't really do the AWA (I only write a quick sentence in order to be able to skip and move forward). Does the calculated score take into account the AWA part or no (in the practice tests)? ie: could I have scored higher had I solved the AWA section?

The AWA is NOT factored into the 200-800 score scale.

But I will say this. Many students tell me that on the actual test in the verbal section they were really tired, and had some focus issues even though during their most recent practice tests they didn't.

The first thing I ask him them is whether they write the AWAs during the practice tests. Because you've gotta realize that that extra hour of mental exertion can really drain from your stamina on test day. So if even for just endurance reasons, we should write the AWAs during the practice tests.

Also, I am aware that GMatPrep is the most accurate indicator for the types of quests I should exam on the actual, but is it also a good indicator to the score I will be getting? My target is a 680:(

The GMAT prep softwre is the best available indicator of the types of questions you will get.

Is it the best available predictor of test day performance? Yes.

Is it a perfect predictor of your score? No.

So, does a GMATPrep score necessarily lock you in into a certain score range beyond any doubt? No.

IF you write the GMATPrep tests under timed coditions (and with the AWAs), and if you a) make the practice-testing conditions as much like the real thing as possible, and if you b) write as close in time as possible to the real thing, then, yes, GMATPrep software must be the best available predictor of your actual test score. This is not to say that the prediction is perfect; there is a magin of error (around +/- 30).

But keep in mind that the margin of error is based on aggregate data, and that an individual, given time and motivation can see big score jumps during practice.