Hello Aspirant,
No, MCC is only responsible for All India Quota counselling and All India Quota counselling is done by DGHS on the behalf of MCC. For state quota counselling there are different state authorities like Directorate of Technical Education in some states, some state universities, and some other authorities. Since you have not mentioned the name of your state so it's quite difficult to guide you with an accurate information.
For more details on NEET Counselling kindly refer to the link given below :-
https://medicine.careers360.com/articles/neet-counselling
To have a list of all those colleges which you can get based upon your marks, category, and state you can use NEET College Predictor:-
https://medicine.careers360.com/neet-college-predictor?icn=QnA&ici=qna_answer
I hope this information helps you.
Good Luck!!
Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile