JEE

Yes, you may get admission in NIT Uttarakhand with a 10328 ST rank in JEE Mains, but in the Special Round. The closing rank for the National Institute of Technology, Uttarakhand, in 2021 for most of the courses was under 10,000 for almost every branch of BTech.
Therefore, based on previous year's trends, you may have some chances to get admission in branches like electronics and communication engineering, civil engineering, and mechanical engineering in a special round. The accepted rank for admission in a special round for the BTech in Electronics and Communication Engineering branch was around one lakh.
However, you may not get admission during regular rounds.

ICT Mumbai Courses, Fee, Eligibility Criteria
Degree Name
Fees
Eligibility Criteria
B.Tech
Rs 3.41 Lacs-
JEE Main or MH-CET scores
B.Pharma
Rs 3.41 Lacs
MH-CET score
M.Tech (Bioprocess Technology/ Food Biotechnology/ Pharmaceutical Biotechnology)
Rs 1.42 Lacs
B Chemical Engineering/ B Pharmacy; B.Tech/ B.E in Chemical Engineering/Food Engineering and Technology/ Food Engineering/ Food Technology/ Chemical Technology/ Food Process Technology/ Food Process Engineering/ Dairy Technology/ Biochemical Engineering/ Biotechnology/ Biomedical engineering/Pharmaceutical Technology/Industrial Biotechnology, or an equivalent course (four-year duration) degree based on specialisation from any UGC recognised university. +
HSSC/ HSC/standard XII with 60% marks (in aggregate) or an equivalent CGPA (5% relaxation for backward candidates). +
Graduate Aptitude Test-Biotechnology (GAT-B). Master of Chemical Engineering/ M.Tech. /M.Pharma
Rs 1.42 - 1.82 Lacs
B.Tech (Textile Chemistry/ Textile Processing), B. Text (Textile Chemistry B.Sc. (Tech) (Textile Processing /Chemistry), BE (Textile Technology or Textile Chemistry B.Tech (Textile Chemistry or Textile Technology), B.Tech (Fibre Technology/ Fibres and Textile Processing Technology) emphasis on chemical processing of textiles.

Based on your percentile, it is not possible to get a seat in any NIT or iiit. Even obtaining a seat in the new iiits via CSAB special round counseling seems very hard. Now coming to other government colleges, you can apply to the colleges accepting admission through JEE Mains. As I have heard, maybe UPTU is planning to take admission by JEE Mains. Look to it, this is not confirmed. Apply to other state colleges accepting Mains result. Give other exams too, I am sure you can get good rank in state exams. Or the other option left for you is to prepare again for JEE and do better this time, you will atleast need a percentile near to 97.5 percentile and above to secure a seat in central colleges considering your category. Try and choose wisely if you are going to take admission this year, do not go for Mains counseling, it would be a waste of time.