You need to change the name on your passport to match your married name. All of your documents (passport, driver's license, social security card) must have the same name on them. You will run into a myriad of problems if you do not get all of your documents to state your same married name. I knew of one woman who had a social security card and license that did not match after her marriage and she was denied a job because of it. It can create all sorts of problems.

In the US, all you need to do is to provide a copy of your marriage license to have a new passport printed with your married name on it. The process should be essentially the same in India, I am sure.

If you do not have a copy of your marriage license, you can get them online at a company called VitalChek. They are online at www.vitalchek.com. They charge you a handling fee of like $25 or so, as they are private company, but they get your license back to you quickly. If you want to avoid their extra fee, you can usually get the license through the mail from whatever vital records office has your license. That process is usually a little cheaper, but takes longer.

If for some reason you absolutely MUST travel out of the country without doing all of this first, I think you should at the very least (1) contact the Justice Department and/or Immigration and inform them of your issue, and (2) bring a certified true copy of your marriage license with you when you travel. Still, I would not travel without getting the new passport if I could POSSIBLY avoid it if I were you.