Well brother, I can understand your situation. I know you have worked hard for JEE mains but I believe with 77.8k category rank you won't be getting decent college which I can recommend you.

Though with EWS and home state quota, there might be an increased chance of your's getting a seat but with such a rank it is very hard to get.

As, opting a very low rank NIT core branch will bring nothing good for you.

So, I would suggest you to prefer state engineering colleges. Some state eng. colleges are way better than even mediocre ranked NIT.

Also, I would recommend you to do take part in JoSAA and go to through all the round even the spot round. May be, some good fortune can turn your destiny. 

Or the better recommendation can be

To prefer Thapur. It is one of the older colleges in education industry with better Alumni than many of the NIT's. 

Thank You