Hey, look Jee mains is a prime examination for engineering students but not the only one. There are numerous other examinations conducted by different states for engineering students. So just register for the ones in which you can now as it's a bit late but there is still hope. Moreover you might have heard of the GFTI which is a government funded technical institute, for example BIT Mesra. And there are quite other good options out there. So I would suggest you to explore your options according to your scores in the individual examinations of the various states. Since you queried for the govt institute, so these are the probable options. But in case you change your mind for private then there are a plenty of options but first go through the state exams.

Best of luck for your vibrant future!!