Maddie,
Mike and Elgin gave you good info so I won't repeat it but I will build on it more and give you some additional info.
If you decide to leave now or wait until your contract is up and they give you a pink slip you will need to go to the local labor board and file it, after that you can usually call it in every week. If you do this have the state automatically deduct your taxes, it will make your life easier when it comes time for you to file your taxes. When you're on unemployment they may call you anytime and ask to see if you've been actively seeking a job so keep track of all the places you've sent your resume and/or visited and filled out employment applications. You can also go to the labor board and describe what your situation is and they can give you all the options you have. They may also help you with locating another job if you wish. I don't know your situation with being between New York and Washington so this may factor in also.
Find out more info before accepting to continue your current health benefits. Health insurance is very expensive and continues to rise. If you need to pay for coverage completely out of your pocket your company may allow you other alternatives like
COBRA which is good for people who need coverage for a short amount of time, typically between jobs. My company allows people to go to COBRA if they're leaving since it's a lot cheaper than staying with Aetna which the company automatically gives us while we're employed there. One word of caution though, some insurance companys will not accept pre-existing conditions, which means if you're currently going to the doctor or other specialist they may not cover you. In that case you should elect to stay with your current carrier. You may want to ask the insurance carriers if you have any questions.
Does you company have an HR person you can go to who can answer some of your questions? They can tell you if you can keep your benefits or not.
As for my opinion, I also say you should stay on as a contractor for now but start looking for another job. In 3 months when they say they no longer need you, you will at least know what's out there if you don't have another job lined up by then. This will also buy you some time in case there's a slim possibility of another position opening up at your current company.