Racheeta,
I read at least a couple of threads related to H1b and jobs. I understand its tough out there.
First, many government agencies do not sponsor a working visa, if you find one that does, you are lucky. These days, even companies are hesitating to sponsor working visas. Even if you could find one that does H1B for foreign nationals, it will be very very tough to get a permanent residency through that job, if that is what you would want after few years. Public agencies have to follow their rules and regulations in hiring and they have to make your case indicating that they could not find an American citizen or a permanent resident to fill the position you would be working.
Having said that, I dont think you should worry about getting laid off even before starting a job. Get a job, work hard and you will be fine. I strongly suggest you to find a job in area that you studied rather than getting a training from some consultancies in area that you have never been exposed or studied before. It may be very difficult for you to succeed in that area if you just have a few months of traning. You probably have spent at least 4-6 years of education in learning something, so you will be better off staying in same field.
If you are seeking some consultancies in your field of study, my response may not apply to you. And lastly, as per my knowledge, there is no such thing such as grace period for H1b holders to find another job. Technically, you are out of status once you are out of payroll, however, attorneys can make a case for you if you can find a job within a month or so. I have seen cases where guys have transferred h1b even after one month of getting laid off. Good luck!!!
