Always make your way to an IIT if you have got choice. Do not compromise the branch of study for the college which you take! And remember the tag 'IIT' is always breater than 'NIT'....

Chitra with 6000 rank in the JEE Advanced examination under the general category you will have lots of choices among the branches and IITs. Although with this rank getting any of the good branches in any of the seven old IITs will be extremely difficult but you will still have a good chance to capture a seat in the new IITs like IIT Indore, IIT Hyderabad, IIT Ropar etc, and further better branches in ultra new IITs like IIT Jammu, IIT Trichy, IIT Goa etc. 

If you have a very good JEE Mains rank which can fetch you a seat like a Computer science, ECE etc. branch in top colleges like NIT Trichy, NIT Warangal etc. and you want to give preference to branch instead of a college then you can look forward for taking admission in NITs otherwise you have a bundle of options in front of you to go in IITs.

You can check out the last-year cut-offs of different branches of different IITs and NITs by going through the links mentioned below:

• https://engineering.careers360.com/articles/jee-advanced-cutoff-for-iits
• https://www.entrancecorner.com/engineering/jee-main-2017-cutoff/

I will personally recommend you to make a right decision, what you want to do either you are good with a brand with a lower branch or you are ready to leave brand and go for the branch.

Feel free to ask if you have any more queries. All the best!!