I've been lurking on this thread, since I have exactly the same starting issues described herein... at this point my plan is to try replacing the solenoid, but, given that the boat is 27 years old, also to re-run all the wiring (and maybe connect the engine stuff to NMEA2000 so I can see if without going to the cockpit.) So thank you all for the thread.
I am writing here for a different reason: the clamp meters (inductive amp clamps) that are mentioned... yeah, baby, I'd buy one of those... but not for this. Or, more specifically, the 'inductive' part is the issue. The clamps just have jaws that close around the wire, and read the current without you having to break the wire and connect it through an ammeter. The problem is that the effect that allows this used to work is based on induction -- basically, the same way your electric toothbrush or iPhone (or induction stovetop) do their contactless charging/heating these days. Unfortunately, that only works for AC currents that change (usually pretty regularly) in time... and the solenoid I assume is semi-constant DC (or, at least, not 60Hz AC like most inductive clamps will be designed for, being household current.)
You _can_ get a DC current clamp, but it has to work on a different physical effect, the Hall effect, that can read at DC. The details don't matter for this forum, but if you're buying, make sure you buy a clamp that has some chance of working for your application. I think that would be a DC Hall-effect clamp. I doubt Harrbor Freight is the place to go for those. Anyway, YMMV but FYI. And Caveat Emptor -- I am not a Hall-effect current probe user, just aware of the issue with the AC (inductive, up until pretty recently (and maybe still) the most standard type) clamps.
C.f.:
https://www.fluke.com/en-us/learn/blog/clamps/inside-hall-effect-clamp-meters ... and yes, Fluke is where I get my DVMs and related tools; in my experience they make 'em tough and accurate, and I trust them as do many engineers/scientists.