S.H.
Do whatever it takes not to have a lapse in coverage or any condition you or a family member on that policy has can now be considered "pre-existing" and uncovered by the new policy.
In regards to the new info you added - they didn't give you "proper" notice so it is what it is - tell them you are sorry but you are desperate and hope they can understand that, in the economy and under these circumstances, you have to take what you can get when you can get it! Take the job and move on!