If it was me I would use the old computer with the serial port for the machine to send data to
while using the new machine for coding and programing.

If you do it that way then one of the Female to Female cables should be fine. Cut it in half, plug into the computer and the wires can go to the bread board.

You will have to use an ohm meter to ring out which wire is which and you will probably have to solder thicker wires to the cable to work with the bread board.

You will only need, in most cases, the wires coming from the DB9 connector, 2, 3, and 5.