Have you looked into buying the COBRA coverage to bridge the gap? If you only need a couple of months that would provide good continuity (you'd only have to go through 1 insurance switch instead of 2) and you wouldn't have the stress of finding a new provider.
I can't tell you how much to expect to pay but self pay coverage is always pricey. A couple of tips: make sure everyone has their dental and eye exams done right before and then let dental and vision coverage go for those two months. Also, if anyone is taking any prescription medication, see if your doctor will right a 90 day prescription if possible and fill it before and then you can worry less about the drug portion of the insurance as well. If you don't feel comfortable letting these coverages go altogether, you may be able to opt for a "high deductible" option that will carry a lower monthly payment.
Good luck!