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Selinoids normally engage or they don't. Nothing half way.
I have to respectfully disagree, but we may be talking slightly different things. I'm presuming the solenoid closes the circuit and pushes the starter gear forward to mesh with the flywheel. I mean that the circuit may close, but may not sufficiently mesh the starter gear with the flywheel gear. Either way, the solenoid is good; something else is the problem.
In an odd-ball case where the professional diagnostic was a new starter, the starter was replaced and situation was not cured. It ultimately turned out to be, (after a lot of head scratching by 3 mechanics) that the starter was getting poor current due to bad wiring; it was closing but not fully engaging the starter gear. The problem finally revealed itself after the battery wire fell out of the end lug. Reworking the wiring cured the problem.
As a side note, the wackiest thing that happened during this ordeal was that the starter motor would kick on and continue to run (grinding) even with taking the key out of the ignition.
Since the op mentioned that shifting the transmission sometimes results in temporarily curing the problem, I'm also wondering if the neutral safety switch might not be operating up to snuff.
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