Done With College, Wanting To Move But Need A Job First!?

I’m on the west coast looking to move to the east coast. I want to live in a big city, my family is in NY, I want to try Chicago, Philadelphia or Boston. I have a degree in engineering design. Throughout college, I worked full time as a auto technician at a car dealership, I have 5 years experience (being only 23, I’ve been told that’s very impressive). I’m looking to work as a CAD drafter or engineering technician. If I apply like crazy what are the odds of finding a job? I’m moving at my own expense and will let the company know that in the cover letter. I hear it’s verrrry difficult to find a job in another state. What are my odds?
And if I was lucky enough to find a job, how much time do employers give for entry level applicants? Is 3-4 weeks reasonable?