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Have you checked the negative battery cable and terminal? This is commonly overlooked and I have seen this problem before.
What is the condition of the cable? You will need to perform a voltage drop test to find out if you have good ground or not.
There are only a couple of things that can cause this issue. You obviously are getting power to your solenoid. So you are either not getting proper power or ground from your battery. Have you tried to jump start it?