Skip white distributor wiring

With the key off, there should be no power to the HEI distributor (IGN1 in diagram). It sounds like you powered your HEI from your coil+ terminal, which is correct. That gets powered via the "wire where ballast was". That is all correct. Measure the voltage from coil+ to gnd with key off and insure 0 V. If not, your key switch is bad or a short in the wiring or bulkhead. My guess is something unrelated, like a trunk or glovebox light is on. Another possibility is the alternator leaks. There is no switch or relay that isolates the alternator output, just the diodes blocking reverse flow (have some leakage). Batteries can also develop an internal short or external flow from dirt and water on top of the case.

Two small points:
HEI works better with an E-core coil.
You didn't have to use a wire to connect the 2 ballast connectors together, there is a small male-male spade you can find in bubble packs.