I want to know how I can change my project within TCS? In my current project the culture is a very toxic lot of politics and no learning.

Stop working and they will release you. Thats the only way you can come out.

Get a ligit medical certificate and apply for 1-2 month LWP. I have done this. I got a release

Does this really work?

Try to reduce your dependency on the project first and they slowly start asking for release rather than destroying your relationship with the team. And try to convince them for release they will agree for it until you don't cause any trouble for them bcoz this what I also did

Take a release citing some other reasons. Its going to take a few months and regular followups. You might endup making a few enemies but thats part and parcel of release process.