While it is possible for a phone to not be capable of running on another carrier's/service provider's network, that itself seems kind of fishy. Sprint and Verizon are both CDMA carriers. So one's phones should work on the other's network. It sounds to me a lot like the person was trying to get out of giving you what you need, to try and talk you into having something else. Through them. Not shocking. They are sales people in the stores.
You will need a SIM through/from Sprint, but there is no reason you would need a newer phone. You may need to talk to someone else at the store (if you go back) even perhaps a manager, or just get the SIM and try setting it up yourself with them over the phone.