@AndyMcKenna No, the way you have the switches wired will not work correctly. If you look at your new switches there should have 3 terminal screws. There should be two copper looking screw(traveler wires), setting across from one another, and one dark screw(power in and power to the fixture) at the bottom of the switch by itself.
Warning: You'll be working around live electrical wires during this procedure. If you don't feel comfortable doing so, please contact a local licensed Electrician.
For this example I am going to use the wire color you used in your picture.
- You show the white/neutral wire going straight to the light this is correct. FYI the white/neutral wires could go through the switch boxes so do not mix the wires up.
- Determine which switch has power coming in from the panel, you can do this with a Non-contact voltage tester or a Voltage Meter. Try to identify the wires as you go with tape or wire numbers.
- After you have identified the power from the panel turn off the breaker.
- At Switch #1, where the power comes in from the panel. This is your home-run switch and you terminate the Black wire on the darker screw. This is your power in from the panel. The Red wire terminates to one of copper screw and the White wire terminates on the other copper screw. These wires are your traveler wires, they lead to the other switch.
- At Switch #2 The Red wire terminates to one of copper screw and the White wire terminates on the other copper screw. The Black wire on the darker screw. This is your power to your fixture.
As for adding switches, unless the power is at the box you want to add to, you will be just adding a switched switch, meaning that until the switch gets power it will not work.