I live in Johnson County Ks and recently have become unemployed with no real opportunities to find a job quick (recent college grad) and rather than not pay my rent and be evicted I informed by Apt Manager I would like to break my lease. However, they are demanding full payment of remaining months of rent – which of course I do not have. Do I have any recourse or relief methods – without screwing up my credit?

You could try to find someone else to rent the apartment. If you cannot, it seems that the law states the landlord will be able to bill you for the remaining months:
Breaking A Lease in Missouri - Labor Law Talk Blog (http://blog.laborlawtalk.com/2007/12/27/breaking-a-lease-in-missouri/)

No, merely being unemployed is not a reason to break a lease.

So basically you are going to break it one way or another, if they evict you, or if you move out ( where would you go) and just not pay, they will end up taking you to court