Why do I keep blowing out my CH4x4 switch?

Use one power trigger for the light and one wire will be power to the switch, and another wire will go to the relay.

 

Usually that's the way it's done, one fused wire feeding the relay and a fused 12V source feeding the switch(or bank of switches if they're all low current).

Obviously the switch is passing current, and more than a little. The small push switch has 4 connections:
One that feeds the switch itself.
One that is the switch output.
One that feeds the courtesy light led.
One that is ground for both the courtesy light led and switch-on led.

The only two that matter as far as the switch failing are switch power in and switch power out, the other two are a separate circuit in the switch housing. Power can only flow through the switch-on LED if the ground wire is connected to either chassis ground or the dimmer ground. I've read of occaisional issues with the button being finicky about turning off or on but one would know it's still on since the lights would be on.

I'd start over using a relay and since op wants the courtesy light wire that up too.