First inclination is an insecure "run" wire at the new ignition, a loose main chassis or ignition circuit ground wire, or a faulty replacement ignition. Check also the integrity of the ignition circuit breaker connections and the main 12v system solenoid, low in the electric bay, streetside front bay.
In fact, that solenoid's history on this Forum makes it a very likely culprit in your case, and a candidate for you to replace or upgrade to an electronic version.
Joel