Dear Monu, 

With 586 marks you definately have a chance for government medical college in rajasthan according to previous years cut off. Also your obc caste is like icing on the cream which makes your Chances of getting government college even more since last year's cut off was 560 marks.

Also you should apply in AIQ for government college and as far as choice are concerned, it depends on which state you want to go.You can also apply for rajasthan from AIQ but chances through state quota will be much more.Just don't fill top colleges in the list...fill according to your marks and what you can estimate to get.

This is a college predictor link for your reference. ALL THE BEST FOR  YOUR COUNSELLING 

https://medicine.careers360.com/neet-college-predictor