What should I do: get an M. Tech. degree at an NIT or IIEST Shibpur or get a job in TCS?

You should avoid going to any NIT to pursue an M.Tech degree. I am saying with this full confidence after seeing my sister’s experience at an NIT. The degree will be of not much use until and unless you want to go further into academics. 

You can join TCS and further prepare for GATE. It would be tough and you might also have to make a few sacrifices but it is not impossible. As one of my friends herself cracked GATE while she was working and now pursuing Master’s in IIT. 

Therefore, do not go for an M.Tech degree either from NIT or IIEST. It would be better that you join TCS and prepare for GATE, so you can get admission to IIT. You can earn while you learn at TCS.