The Ford V10 has 2 starter solenoids but both are not shown in their diagrams. We had these exact same symptoms until one time it would not start no matter how many times we tried. At that point, we had already replaced the solenoid up front but had read about one that is accessed either thru the doghouse, or by turning the wheel to get behind the front passenger tire area. Tried the doghouse first - arms were not long enough. Husband finally got to it behind the tire. No more problems. It was intermittent at first, then became - how many tries before it starts this time gradually taking more and more tries. Sounds like the same thing.
Ours is a 2000 Monaco