Hello,
As you mentioned with SC category and 377 marks, your chances of getting a decent government college in your state through state quota(85%) are quite high.
As the admission to medical colleges based on NEET is given under All India Quota(15%) and state quota(85%).
It depends very much on your state.
However cutoffs change every year depending upon level of toughness of exam. So hope for the best and follow the counselling process.

Follow this link to better predict your college-
https://medicine.careers360.com/articles/neet-college-predictor/amp

Good luck

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