Sounds like a liability issue from the vendor. They don't want to take a chance that the customer will use a 3 wire cord. And even if they do the vendor can say that hey they bought the right cord.
The "right" cord is the one that fits the receptacle, whether it is 3-wire or 4-wire.
Existing 3-wire circuits, receptacles and cords are not "illegal" to keep and use. If you get a new dryer you certainly CAN put a 3-wire cord on it and keep using the existing circuit.
The sales man said that a new cord was required, whether 3 or 4 prong, because old cords go bad and cause electrical problems.
Maybe they do, but there is NO requirement that you replace it.
Actually the receptacle is what usually fails, it just takes the cord with it.
What they meant was that code requires a 4-wire cord. "new" in this case meaning 4-wire as opposed to an "old" 3-wire cord.
Just for clarity, more modern codes require a 4-wire circuit and receptacle. The cord just goes along with it.
But I know we all knew this already.