Exam

NMIMS does offer an integrated MBA program, but that is only for the family business specialization.
Kindly go through the below link for more details.
http://www.nmims.edu/admissions/

XLRI is the best college which you can get through XAT.

Unfortunately, with that score, it is impossible to get a call from even a decent B-school. I suggest you to opt for a retake (if possible) or focus on other upcoming exams like SNAP, CMAT and CET. Good luck.

You should take IELTS.

You have a fair chance of getting a call for the WAT/PI process of VGSoM on basis of the 2017-19 selection criteria. If you have some work experience, it will improve your chances further.
Final selection into the program depends on you scores in WAT/PI coupled with your Application Rating (AR) score