Well a 560 is already above approx 50% of people that take the GMAT, so this is not a terrible score! This score would probably keep you from gettting into Top FULL TIME programs, but since you are planning on going part time you can in fact have a shot at some good schools. Part time is much less selective.
The best school within your score range, and within your distance is USC.PM which has an average GMAT of 620 and an 80% range of 540-690. So your score would be on the lower end, so you'd have to bring something special somewhere else, but your GPA is much higher than the average GPA there which is only 3.11. USC.PM also accepts approximately 75% of the students that apply, but only about 60% of those accepted end up matriculating. Obviously, this school would be much more expensive, but would also have a better reputation than USD. The challenge for you would be that the classes are offered at nights. You should check into USC's satellite capus in Orange County. They offer the same classes in Orange County and you still will receive the USC degree, and this would be much closer to San Diego for you.
The best MBA program in in the SD area I believe would be the school that you already applied to and UCSD (if they have MBA program), but there is also Point Loma and CSUSM. I believe you'd have a great shot of getting into all of these programs. Also, you'd have an excellent shot at every other school I mentioned earlier. The only school that would most likely be out of reach for you would be UCLA's FEMBA program b/c the average GMAT is 690, and you would not be within the 80% range.
PS, you are correct about weight on the Quant section. A 32% quant will not do you any favors. Are you ready to start an MBA program which will be very quant intensive? To mitigate a low quant score you will have to show usage of quant skills in other areas of your life or profession. You could also take a stats or calculus class at a local college and get an A to help.
How long did you study for the GMAT Quant? With your strong verbal, you'd be able to easily hit much higher scores with some improvement in Quant. It's much easier to improve quant than verbal. If you were able to hit the 60-70% in quant, i believe this would be enough to mitigate any concern over your Quant background at the schools you are targeting.
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