This is an update to this post.
When I transfered my bank account from Coimbatore to Chennai, I asked the Coimbatore branch to transfer my account to a Chennai branch that was around 3-4 KM from my house. But after I came here, I noticed that there was another branch (of the same bank) two streets behind my house!
So, I applied to the initial branch to shift my account to the new branch (behind my house) and update my new address. They transfered my account, but did not change the address. Maybe they thought I was going to a different branch anyway and I could always get it changed there. I wrote the earlier post when I came back on that day.
The next day, I went to the new branch and requested them to update my address. I submitted my ICICI bank credit card statement and a copy of my post office PPF pass book (which had my new address). They asked me to submit it and go.
I went back today to see that my address has been updated in the bank. They have printed out my new address in one of the pages in my pass book as well (they will give a new pass book only if this one gets over).
I think that should be sufficient and will be accepted by other departments as proof of address. Or, I need to make a lot of small transactions and keep requesting for statement printouts till all the pages in the old pass book get over. Then I will get a new passbook with my new address printed out in the first page.