I'm betting the "new switch" you got off eBay is either the wrong switch for your application, or simply not good. $39 for a new one including shipping sounds fishy. A new switch should cost well over $100 plus shipping.
I had the same problem on my '77. Get a good switch made for your car. Even though the 280Z's appear to be the same, there are some differences in parts.
As you mentioned all your lights work from your alarm, and to agree with T-Bone, it sounds like your light circuit is fine (power, ground and fuse). Something is funky with your switch. If you can't fix your original switch, I'd get a good used one that is from the SAME year as your car ('77 and '78 are the same).