Any moisture issues related to the standing water incident would have manifested itself long before now. Whatever the current issue is, it's unrelated to the previous issue. In that time, it is possible a new leak developed. It's also possible the dark areas are related to interior humidity and unrelated to the roof. In cold weather, the ceiling surface under rafters is measurably cooler than the nearby portion under insulation alone. This can cause condensation only under the rafters. As you know, condensation can lead to mold.
Can you tell if the darkening is due to actual mold? The damp surface can attract dirt that darkens the area even though no mold is growing. Even though the dirt can stain, attempting to clean an area with plain soap and water would yield a very different result than if mold is actively growing.
If mold is actively growing, it is time to consult with a mold professional. They will determine the cause, mitigate the mold, and suggest how to prevent it from happening again. If you want to investigate further before going that route, the only reliable way to determine if the rafters are getting wet is direct inspection.
Cut a small inspection hole where you think it's the worst. If it's wet up there, a lot more ceiling is coming off, so don't bother patching the hole. The roof is leaking. It needs to be repaired and all wet drywall and insulation needs to be replaced. If it's all dry, breathe a sigh of relief. The need to patch the hole is much less than it could have been. Be sure to adequately seal the damaged vapor barrier.
If the rafters are dry, the only thing you can do to reduce condensation on the ceiling is reduce the overall humidity of the house. First try selective ventilation and lifestyle adjustments. Consider bumping the thermostat up a bit. If that's not enough, look into continuous ventilation through an air to air heat exchanger.
A far as replacing stringers, that cannot be done with comp roofing unless a full cold roof is installed. It is normal practice for concealed structural spaces filled with insulation to not be ventilated. It sounds like the roofers followed conventional building practice.
That's at the PEAK of a shed roof, true? If that's the case, what you have amounts to a ridge cap, but nobody I know of manufactures premade ridge caps for shed roofs (too many different possibilities, not enough market), so the builder used "drip edge" or flat flashing (bent around the peak) to function as a ridge cap.
There really aren't very many good alternatives to that situation, so what they did is actually pretty common on the peaks of shed roofs. They really SHOULD have tarred over the nail heads, though.
Best Answer
According to GAF Roofing manufacturer’s installation guide, the shingles should overhang the rake between 1/4” and 3/4”.
Here’s their website that explains it better: https://www.gaf.com/en-us/document-library/documents/residentialroofing/shingles/glenwood/glenwood_shingles_application_instructions_english_spanish_french.pdf
Yes, you should be able to push the shingles up about 1”, but if they extend too far beyond the edge of the roof (especially in a high wind area like you live in) they’re more susceptible to blow off.
Also critical is the correct number of nails per shingle. In high wind areas, 6 nails are required.
GAF is a major Roofing manufacturer in the U.S., but I’d check with your shingle manufacturer’s instructions.