Hi nawaf52,
Based on the information that you provided in this post, there are a number of issues to discuss:
1) Test Day is a rather specific 'event' - the details are specific and they matter, so you have to train as best as you can for all of them. The more realistic you can make your CATs, the more likely the score results are to be accurate. The more you deviate, the more "inflated" your practice scores can become - and that's likely what has happened here. By skipping sections, you took shorter, easier Exams that required less effort than what you'll need for the Official GMAT. If you did anything else that was unrealistic (such as taking the CAT at home, taking it at different time of day from when you'll take the Official GMAT, etc. then the score results could be even more inflated. This is meant to say that you are probably not at the mid-500s ability level right now.
2) If you were actually at the mid-500s level, then you would likely need at least another 2 months of consistent, guided study to hit 650+. Considering how you took your CATs though, you will likely require at least 3 months of that type of study. With your current Test Date, you have just over 3 weeks of study time remaining - so you might want to consider pushing back your Test Date.
3) The GMAT is a consistent, predictable Exam, so you CAN train to score at a much higher level. However, 'your way' of approaching the Test hasn't gotten you close to your goal - so you might want to consider that some expert 'guidance' will be required. The amount of money that you spent taking the GMAT those prior 4 times would have been enough to pay for a GMAT Course, the guidance that you need AND the fee to take the GMAT (with some money left over). Considering the magnitude of what having a PhD could mean, wanting to avoid spending money on a GMAT Course right now doesn't make much sense.
PhD Programs tend to be highly competitive, since there are so few spots and all of the applicants tend to be really qualified. As such, you can expect that all of the applications will be 'nitpicked' - and you don't want your application to have any perceived 'weak spots' (such as a lower GMAT Score).
1) What are the application deadlines for the Programs that you're interested in?
2) Going forward, how many hours do you think you can study each week?
GMAT assassins aren't born, they're made,
Rich