Which iiit will I get with a rank 0f 56377 in JEE Mains 2019? (though for my marks should have got nitk://). Will I get IIIT dharwad?

The East Point Group of Institutions accepts many exams for the BE course. The accepted exams are JEE Main, COMEDK, and KCET. Students can appear in any of these exams to get admission at EPGI. Students need to fulfil the eligibility criteria as well.

Amrut Institute offer admission in BTech and Diploma in Engineering based on merit in last qualifying exam. So, no JEE Main is not compulsory to take admission in Amrut Institute courses. 

With 57.8 percentile in JEE Main and 73% boards, getting CSE/AI in top government colleges is unlikely. But you still have good chances in private colleges through UPTAC/AKTU or direct admission.

Possible options:

KIET Group of Institutions

ABES Engineering College

Galgotias Coll

JEE (Main) cutoff for BTech in Computer Science and Engineering in Sagar Institute of Science Technology and Research, Bhopal in the closing round has ranged between 3,75,232 to 10,07,060 from 2021 to 2023.

Yes, you can get admission in BTech in Sagar Institute of Science Technology and Research without JEE (Mains) rank through College Level Counselling based on merit if there are any seats left vacant after DTE Bhopal Counselling.