Hi Xenion7,
There are a number of important factors in your situation, which I'll go into. First though, the immediate answer to your question is probably NO - you're not likely to improve 270 points in 2 months. However, you could possibly improve that much with MORE time and the right combination of resources.
I want to review some of the information that you've provided, since it impacts your situation and potential:
1) You basically did no practice before taking the GMAT. It doesn't sound like you had taken many (if any) practice CATs before this Test, so you were unprepared to face the Official GMAT. The average score on the Official GMAT most years is right around 540-550 (and that is a large group of people; most put in some serious study before taking the Test). We have no idea of your "peak" score is yet. This score is just a "starting score."
2) That prior point having been said, you have a LOT of work to do to raise those scores. You'll need to learn (or relearn) the content as well as the tactics, patterns and 'secrets' to the Test. Most Test Takers spend 3 months (or more) of consistent study time preparing for the GMAT; during that time, it's common to spend 10-15 hours per week (or more) on the process.
3) There are some non-content areas that you also need to work on. The pacing and 'energy' problems that you faced in the Verbal section have to be dealt with. If you did NOT answer those last 3 Verbal questions, then they were automatically marked as Incorrect AND were then PENALIZED. Going forward, you have to make sure to answer EVERY question, even if you're just guessing.
4) You mentioned that the MEAN score at the School in France is 650. That does NOT mean that you need a 650 - it means that you need to be "in range" of what that School considers an acceptable GMAT score. A 550 would probably be too low, but since neither you, nor I, work for that School, we can't say that for sure (especially if the rest of your application is strong).
You don't have to make any decisions about all of this just yet, but given your goal (and the 380 that you're starting with), you're likely going to need to invest in a GMAT Course of some type (either Guided Self-Study or Instructor-Led). Most Test Takers can't make huge score leaps on their own.
When do you have to decide about whether you'll attend the School in Spain (or not)?
GMAT assassins aren't born, they're made,
Rich