Usually you can't collect unemployment benefits if you left willingly. Generally they are available only to workers who were laid off or let go on good terms. If you are involved in a DV program, there might be some way around this, because fleeing for your safety would certainly fall under 'just cause' for quitting your last job. (I would leave out the part about moving because the economy is better.) I would apply and see what happens, knowing you may need to provide some personal details and/or appeal their initial decision. Also look into other programs you may be eligible for -- sometimes there are grants/financial assistance for those in DV programs to help them get back on their feet, or you may qualify for emergency food stamps, medical assistance, etc. Bottom line, I'm sure there is some way for you to get some kind of financial help, and UE does sound like a good place to start!
Co-sleeping is really wonderful when your child actually SLEEPS!!