It's the ignition switch. You are losing the ignition signal when in the run position. It will stay running if you hold it in the start position. Make sure you replace the ignition switch on the steering column not the key and tumbler.
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=== Update (09/17/2020) ===
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Did you replace the key and tumbler or the electrical part on the switch?
It's the ignition switch. You are losing the ignition signal when in the run position. It will stay running if you hold it in the start position. Make sure you replace the ignition switch on the steering column not the key and tumbler.