Sounds typical of loose or dirty battery cable connections. Ford has a lot of trouble with loose connections on the large copper solenoid nuts but you would have overcome that when you replaced the solenoid. The best approach is to use a digital voltmeter or test light to check for voltage, starting right on the battery posts, then working your way down the line until you find the spot where voltage is lost. Do that test with the headlights turned on so there will be some current trying to flow. That will make the tests more accurate. You also want the headlights to be not working so the problem is occurring and can be found. If necessary, make the tests while a helper has the ignition switch turned to "crank".
Don't overlook the small black wire from the battery negative cable where it's bolted to the body. Ford has also had a lot of trouble with the large battery cables being corroded away under the insulation where you can't easily see it but that typically occurs down by the starter and doesn't affect other circuits. Testing for voltage at various points will identify if that's the cause.
Wednesday, November 30th, 2011 AT 9:26 PM