I'll try to touch on some points not discussed much by others.
I would explore different areas to live. Health insurance will be a big issue, but under federal HIPAA law (forgetting anything about health care reform) after you use up your COBRA each state has to provide at least one option for you to buy insurance, even if you are otherwise unisurable. NJ is expensive. I have no idea if they have subsidies for low income people and if they do, what they count as income. Look around at other states as the cost varies a lot. You might read our FAQ on buying private health insurance. (FAQ archive) Buying Private Health Insurance
What is not covered in that FAQ is that some states are subsidizing health insurance for those who are uninsurable, if you meet certain income requirements. I don't know if you might qualify for a subsidy, but be sure to look at that possibility as you look at state options. Don't trust what you find on the internet regarding subsidies as there have been many changes in the past months. For example, some have mentioned Texas as a low cost of living state. Texas has an insurance risk pool you can join as you come off of COBRA. Texas may
have a subsidy. http://www.tahp.org/documents/HRI/PCIP%20final.pdf
One problem in figuring this stuff out is that with health care reform there often are two risk pools operating in a state. You are only eligible for the federal pool if you have been unisured for six months.
As you can see, this is confusing. As you poke around you might want to post any questions here. Also, you might consider talking to agents who work in states where you might want to live to see what is available. However, some agents are not very up on the HIPAA and risk pool options and subsidies.
Make a list of places you might be interested in and then see what is available for housing, insurance, etc.
You can move while on UE, but I believe that you will have to file in the new state.
See my signature.