Since you cleaned your own contacts to the lights and got them to work, I won't answer that part of the question. That leaves the dash lights and why they won't work.
The 61 Apache has a switch which looks something like this:
You'll notice on the front of it there's a spring looking kinda thing which encircles the it somewhat. That's a rheostat which, when you rotate the light switch knob, allows you to adjust the brightness of the dash lights. What I believe may have happened to you is this part is dead, not allowing power to flow through to the dash lights. The only cure to fix this is to purchase a new switch. The one pictured is for a '61 C10 truck, which should be (IIRC) the same as the one for your Apache. At the time of this post, the cost from Rockauto.com is ~$55.
Replacement of this part is really simple. The first thing you need to know, there's a button on the bottom of this part (as pictured), which when you press it allows you to pull the knob all the way out. On the dash, you'll find a circular piece the knob sticks through. There should be a couple of slots on it. Spin this counterclockwise to loosen, then remove. (You can see the threaded portion in the very left portion in the metal on the very left side of the image.) This will release the switch from the back side of the dash. Unplug the connector from the top of the switch. Reverse the order putting it back in, though all you have to do is stab the knob back through without pressing the button. You'll see what I mean when you pull it apart.