I must have applied to some 300 jobs, had over 200 calls from the recruiters and the direct employers, who would turn away as soon as they hear that i would need H1B sponsorship,,A company finally agreed to call me in for on site interview for a position that they did not even have as an opening in their website. They called me based on my inquiry for any possible opening in their place and that was my first job,,kind of underpaid one when it started. But it has worked out well since then.. That was some 7 years ago..not sure how it it now..And yeah mine is engineering field too..