Did they charge your credit card? That is key! If they charged your credit card you own the phone. Ask for your tracking number.
Turn your old Sprint phone off and remove the battery. Then power your new iPhone off (after you registered with Apple) then on again. Now your activation will work.
Turn your old Sprint phone off and remove the battery. Register your iPhone then power it off/on. You will now be able to activate. Congratulations on the iPhone!