That 4.5A rating is probably the steady-state current. When switched on the motor will pull stall current (perhaps around ten times as much) and has spot-welded the contacts together.
This is one situation where a MOSFET can be more reliable if its pulsed current rating is upto the surges. Something like 0.01 ohm, 120A rating MOSFET would probably do nicely - even if the motor winding resistance is as low as 0.1 ohm it will handle the surge and dissipate 0.2W steady-state. For ease of interfacing ensure a logic-level MOSFET is easiest or you'll need another transistor to drive it.
And because it is inductive the motor will need a backwards diode across it rated for 10A (pulse) or so to protect the MOSFET from inductive spikes.