Cooper 7503w is a 3-way switch.
It sounds like you purchased correct switch.
Remember, fixya answers questions about products. We don't sell Cooper wiring devices.
You have a continuity tester which is excellent, because electricians test, they don't guess.
Hopefully you have a voltage tester for testing steps shown below images.
Notice that new switch has 1 dark screw and 2 brass screws.
Dark screw is 'common.'
Brass screws are 'travelers.'
Screws are color-coded so electricians know where to connect wires
If your switch is a 4-way, it will have 4 screws: Notice switch located in middle of 3-way switches has 4 screws.
If old switch is 4-way, then each time switch is flipped, it will reverse travelers on either side of device.
The same is true with a 3-way switch. Each time switch is flipped, the travelers are reversed.
So a 4-way switch can be used as a 3-way switch, so old switch might be 4-way switch substituted for 3-way.
The key is testing wires in your box.
You have two switches that control same light. Switch1 and switch2.
Switch1 is still installed. Move switch1 to DOWN position.
You are replacing switch2 and have unknown wires at that location.
Switch1 is DOWN position.
Separate wires for testing.
Turn power ON.
Test each wire against each other wire and against metal box to find Hot wire with power. Mark this wire.
Move switch 1 to UP position, and test wires again to find Hot wire.
If same wire is Hot each time, then that wire connects to dark-colored screw.
If different wire is Hot each time, then these are the travelers and these wires connect to brass colored screws in either order. The final wire connects to dark screw.
Now if you have additional wire, that could be wire going to next receptacle.
Leave this wire disconnected and check nearby boxes for power.
If this is a power wire that advance power to next receptacle, it probably connects to dark screw.
Again, you should test, and use process of elimination until each wire is identified.
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