Hello, 

As you have mentioned that your neet score is 563 marks and you fall under general category. Indeed your marks are good that they would fetch you a seat for government medical colleges. 

Though the round cutoff is 582 but still you have good chances to get a government medical college.

Wait for the further rounds of the counselling.. If you don't get a college till 7th round then its sure that you can get a college in the mop up round. 

To predict the colleges you may please visit the link given below :

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

I hope this would help you. For further queries please feel free to write me down in the comment box.