Hello there! Hope you are doing good.

Yes one does need a MBBS degree to go for anesthesiology.
Some colleges that offer anesthesiology in Tamil Nadu are:-

Madras Medical College, Chennai

Shri Ramachandra Medical College & Research Institute

Sree Balaji Medical College and Hospital

PSG Institute ofMedical Sciences& Research, Coimbatore etc.

Hope you found this answer helpful. Good luck for your future!