Hold on. I think we're mis-communicating due to a problem with terminology. In your title, you listed, "vehicle cranks but does not start". In the first post, you said, "Turn key on dash lights work no crank", then all the tests you listed involved the starter system not working. In your reply from last night, you said, "it wont start anymore but it still cranks." That agrees with your title to this question, so you can see my confusion.
I've been thrown for a loop in the past when someone said they "cranked the engine", meaning they turned the ignition switch to the "crank" position. Then they added, it doesn't start", meaning the starter doesn't spin the engine. The term, "doesn't start" has always been a point of confusion that is often aggravated by radio commercials. To be sure we're talking about the same thing, I use the symptom description "crank / no-start" when describing the starter spins the engine, but the engine doesn't run, or I avoid using "doesn't start", and use "cranks, but doesn't run".
Another one is "turns over". Some people think that means they turned the ignition switch. "Turns over" means the starter is cranking, or turning, the engine. In an attempt to avoid this confusion, some of my replies become somewhat cumbersome or wordy, but it insures I'm thinking of the correct set of symptoms.
After rereading your comments a few times, any time you say the starter cranked the engine, please be sure to make clear whether that was by turning the ignition switch to "crank", or by jumping the two terminals at the starter or by doing something at the starter relay.
Now that I got that out of the way, last night you said, "it wont start anymore but it still cranks". Do you mean the engine is spinning when you turn the ignition switch to "crank", but it won't run? If so, try that while holding the accelerator pedal down 1/4". If that gets the engine running, there's an easy fix for that. I'll describe that later to avoid more confusion now.
Earlier today you said, "nothing on the blue/yellow in the crank position". If the starter engages when you turn the ignition switch to "crank", there has to be 12 volts on that blue / yellow wire. Where I usually make the mistake is not getting the probe in far enough or at the correct angle to touch the terminal.
Rather than wait for your next reply, let me jump ahead as it is starting to sound like the new switch has a problem. I see what you mean about the yellow / dark blue wire. It is in one of the two smaller holes. You might have better luck sticking a straightened-out paper clip in there. You have the correct two terminals, those on each end of the connector. A quick way to verify the switch is at fault is to jump those two wires together right there at the back of connector. You can do that with a pair of paper clips or a piece of wire. The engine should crank, but it won't run as long as the ignition switch is off. If you want the engine to run, turn the ignition switch to the "run" position, but be sure the transmission is in "park". We don't want you to look funny chasing after the vehicle as it drives away.
Oops; it just occurred to me the starter still won't engage unless the neutral safety switch is in the "park" or "neutral" positions. I'm ignoring that entire part of the system for now because your previous tests at the starter relay socket showed that to be working.
If you can't get the starter to engage by jumping those two wires, pull the connector off, then try again by going right on the terminal end with the paper clip or piece of wire. Now if the starter engages, either the new switch is defective or one of those two terminals is spread open too far to make contact with its mate. When a terminal is spread open, you can often get the circuit to work momentarily by tugging on the connector to bend it a little. That can move the spread terminal just enough for it to make contact.
Let me know what happens when you jump those wires.
Tuesday, February 4th, 2020 AT 3:50 PM