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No one is going to tell you this but the fact is that in the next 4 years of your B.Tech, you want to enjoy and study. You do not want to waste those years in trying to do something that you are not even interested in. So to answer your question, if you are passionate about CSE, if you are interested in computers, just do not think twice and go for CSE in some new IIIT. You are getting the branch you want and moreover, it a IIIT. New old, it has a reputation, and moreover, there will be companies there for your placement too. There is no problem with taking a new IIIT at all. On the other hand if you take the NIT with any branch you get, you are gonna be miserable for 4 years. You would not believe able to enjoy your classes. You will keep running from college and exams. You will be a little frustrated. That is the problem. I hope this helps you. All the best.

There is no relation between your family income and B.Tech fees, if you are hard working and intelligent student, you can do best to get admission in India's best college which havevery low fees
If you achieve a good rank in JEE ADVANCE they you will get IIT which is dream of every Science student in India, and you will get upto 100% scholarship in it, that means your B.Tech will be completed at no cost. If you achieve good rank in JEE Mains then you will get India's top NIT and IIIT to pursue B.Tech and yiu will get scholarship also there, so you B.Tech will be at a very low price
You can also prepare well for CUET which is common entrance exam for central university, government University like JMI, BHU, AMU, university of Allahabad, and these universities have very low fees approximately 10,000-20,000 per year, and they also provide scholarship
If you are not enough to get good rank in JEE Mains or CUET then you should take admission in low private colleges, which assure you to provide scholarship. SEE it depends on your hardworking and your preparation, many students are from humble background, and they still prepared well day and night to get admission atleast in government or central University or government college
So it's just excuse that you are from humble background so you can not take admission in these types of colleges and universities which have very low fees.