I know I'm a 5100 user and thus can't really compare. But if you are anything like me, if something is wrong with a product or if a product has a disadvantage that I feel is major then I have a hard time living with it. My suggestion to you is to:
1) Return the Fuji to sears and buy another one elsewhere.
2) As much as I like fuji cameras, ditch the 5200 all together and go with a panasonic
(if you want raw) or if raw doesn't matter go with a Sony or Canon superzoom. Again, I would usggest staying away from most Kodak superzooms as there IQ isn't as good IMO.
3) If not in a hurry wait for the Fuji S6500 as it has the potential to be a really nice camera
I agree though, that the amount of PF you have is severe in both copies of the camera you have tried. I personally wouldn't be okay with living with it.
However, I will caution you that you will not be able to get a P&S (even if a superzoom with raw) that will have near-perfect image quality like a dSLR. To me the best image quality currently on the P&S market is the Fuji F10/F30, but these can have a hard time in outdoor lighting (blowing highlights and such, and I don't remember if they have PF problems or not). But the S6500 will be using the same sensor as these and it could have better results than the F30 model.
Any camera you choose will be a compromise. You need to determine which ones you can live and which compromises you can't.