NIT

Hello Sathvika,
You have to appear for JoSSA 2019 to get the admission in 31 NITs and 23 IIITs and 23 other government-funded technical institutes (GFTIs) and 23 IITs if you will be able to qualify in JEE Advanced.
To appear in JoSSA 2019, you have to register and fill the choices of college that you want and I think registration and choice filling will start from 2nd weak of June in online mode and seats will be allocated based on your marks and your preference of branch and colleges. To know about which college you can get based on your marks in JEE Mains please use college predictor
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
Hope it helps!
Good luck!

There is no common counselling organized by the COA or NATA. Each state carries out its own counselling based on your 12th percentage, NATA score etc. So you will have to do your research, see if the government colleges of the state you're interested to study in have all India quota seats or if you satisfy their domicile requirements etc. It's a very tedious process, you will have to visit the DTE website of that state, read up on all the eligibility criteria, see if any additional exams are required to be answered, check out the application dates etc. At times the websites are not updated and do not provide sufficient information. You will have to call or visit their DTE office to clarify.