I'm not sure if I understand the situation, but it sounds like you were driving it with the new dizzy, and after shutting it down, the car won't start any more? Was the engine cooled off when you tried to start it, or was it still hot?
If the relay is clicking, then it sounds like it might be the starter - is the starter spinning when you turn the key? If your timing is off, you could have a heat problem that's causing thermal soak on the starter. That will cause the relay to click without turning the starter, whenever the engine bay gets too hot.
No matter how you wire the distributor, I don't think there's a way to kill your battery (this will be the part when someone else explains how that happens) so if it turns out to be a power supply problem, I think that might just be a coincidence.