The exhaust elbow (assuming water is coming out stern) would show symptoms overheating. It's possible that some the soot could be from back pressure from a mixing elbow -- but that's not what you'd get from a "sudden heart attack" of your Westerbeke.

o Plugging up of your pick-up tube, blockage in the fuel hose; or collapse of your intake hose -- they used to have a screen on the pick-up tubes, you're may have become plugged-up so it's very constricted?;
ol Fuel pump problem (I wonder why it would be intermittant, but...);
o A kill switch short (if you have a button vs. a pull lever?);or
o A sudden stop would indicate either a complete obstruction of the air intake (unlikely from a little problem);

The first boat that I owned (with a friend of mine) was a new Pearson 35 (Hull #473) -- so it wasn't a new model ;^))). When we took the boat out for the first time, it started like a champ and we motored-out, had a nice sail.

Then we started the engine to motor back in. It ran for a few minutes, then died. I went down below, tried to bleed the engine and other things. After a while, it started. I felt like a hero.

The engine ran for a few minutes, then it died. I did my fussing around. It started then died. I obviously wasn't fixing the problem.

My partner brought us literally 'into the dock' with the 150 genoa in very light air.

After the dealer sent-over the engine mechnanic, they discovered the problem: The pick-up tube was too close to the bottom of the tank and it literally sucked-up the bottom of the tank. After the vacume let go, you could start the engine until it did again.

Have you looked in the tank lately? Does the pick-up tube have a screen on it and is it ocluded? (Take the thing off.) My guess (absent a cut-off misfunction) is that it is a fuel problem, as others have stated before me in this thread.