If you get an iPhone, you need to sign up for a voice and data plan for it (or pay the extra $400 for an unsubsidized/no-contract iPhone). So, your iPhone will have a phone number and have a voice/data plan regardless. You can sign up for a family plan with 2 lines and get a flip-phone for one line, and port your current number to that phone.
If you won't need constant internet (3G) access and can live with Wi-Fi access only, consider an iPod Touch instead of an iPhone.
FWIW, my wife and I switched from flip-phones to the iPhone with no issues. Without delving into your personal reasons, you might try it for a while. The phone is easily locked, slips into a pocket/purse and won't unlock itself, etc. One suggestion - keep your Sprint flip phone for a little while, buy an iPhone and use it for a bit. Make some calls on it. Give the number to one or two people. Then, if you get used to it and can live without flipping out, so to speak, dump the Sprint phone and you can port your number to the iPhone account at that point. Else, change your AT&T account to a family plan at that point, and port your number to the second AT&T (flip) phone.
I don't know of too many people who would
want to carry multiple phones. Personally, I told my employer "no" for a couple of years to the free BlackBerry they told me I needed to carry, and waited until IT started supporting iPhones, to avoid the need to carry two devices.