Hello and welcome to Just Answer!
It sounds like you have an ignition switch issue. The ignition switch controls several circuits individually and when the solenoid won't stay pulled in when you let the key back to the run position, I need you to check fuse #9 for power. This fuse is in the interior fuse box and I need you to leave the fuse in and use a test light to probe the test port on both sides of the fuse with the key still on.
You should have power and if you don't I need you to remove the upper steering column shrouds and test the dark blue wire for power at the ignition switch with the connector still attached to the switch. The dark blue wire must be live in both the start and run position and I think the run position part of the switch is causing the problem. This will also cause the problem with the wait to start light.
Test this out and let me know the result and we can go from there!