I got BS from Korea and MS degree from US.
My major is Mechanical Engineering (Systems and Controls)
I started from OPT and finally got Green Card.
Here are few tips.
Always VISA first. If you want to stay in US, apply for any position sponsoring your H1b or Green Card.
It might be hard to get a job at the very first time, but once you get one, next time will be much easier.
Apply as much as you can. It may be more than a hundred.
Ask for reference if you can get.