The first thing I would suggest is to consider why you are moving? Does the new city offer better opportunities in your field of study? Is the pay higher? Is it a better quality of life for you and your family? If the answer is yes, then it's a viable consideration. If your itchin' to get out just because then applying for every job that seems remotely doable is never the way to go. And please Lawd, don't tell me that you are moving to be with a man or some other silliness. I only bring that up because I've heard stories aplenty about smart, savvy, beautiful women of color chasing down a man with the promise that he would put a ring on it as soon as you step off the plane. Don't fall for that, especially in this day and age where getting a well-paying, meaningful job in a different city has quickly become a commodity.
Now, back to the matter at hand. If you've applied for four jobs and haven't heard a peep of interest from the prospective employer, then why would you think this is a good idea? I don't know your situation, but I do wonder if this is a well thought out plan or a flash in the pan. Before you can even think about moving, ask yourself the following:
1. Why I am moving?
2. Will this move yield better opportunities?
3. Is the company a better one than where I am currently?
4. Is there stability and room for growth within the company and industry?
5. Is the new city a place where I can really call home?
Once you answer those questions, start reaching out to new people in said city for contacts, resources and information. Ask not only about the job opportunities, but also about the cost of living, residential options and overall attitude of the people that live there. Build a good support team around you that includes family, friends, church friends, future colleagues, etc., because these are the people that will keep your strong during the transition. Without them, you could be setting yourself up to fail.
Sometimes we just want to get on that next train smoking, but if there are no realistic job prospects, then I say, stay put, rethink what you are doing and why you want to do it, then jump into action. It's easy to fall in love with a cute condo, 3-story walk-up or 1-acre lot on a cul-de-sac, but it won't make a damn bit of difference if you can't pay the bill.