Welcome to DCI!
Do you have any other device that can take SD cards that you can try 'em out with? Even a card reader would be useful. Since a new card does the same thing as the old, it sounds like it's the camera that's at fault here, but it would be handy to know for sure whether the cards are OK.
It's stating the obvious, but if the cards check out OK, then the camera's had it.
Oh, and for future reference, you don't need to pay through the nose for Kodak brand SD cards. Any SD card from a reputable manufacturer will work just fine in a normally-functioning camera. I would usually list the likes of Crucial, Lexar, Kingston, Sandisk, etc. as reputable, along with a couple of other less well-known brands.
To give you an idea of the price difference and using Amazon as a baseline, for the price you probably paid for that Kodak 512MB card, you could easily have had a 1GB card and quite likely a 2GB card instead.
Heck, it's not like Kodak even makes their own cards anyway. I believe they're just rebranded cards from Lexar, actually.
Oh, and congratulations in advance on your new little baby. I was in the same have-to-make-sure-the-camera-works situation a while back - our first daughter was born just short of seven months ago.
