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I would turn off the heat mat as soon as the majority have sprouted, otherwise they can get leggy. I would also remove the dome at that point, if you leave it on too long after that then it increases the chances of fungal issues like damping off. But with most seeds you'll have the best luck if you wean them off of the dome gradually rather than just taking it away altogether--there's a huge humidity difference between inside the dome and your house in the wintertime, so it's best to adjust them to the lower humidity gradually
Take it off for a short period of time, then put it back on. Later in the day, take it off again for a little bit. Next day take it off for a bit longer, and keep increasing the time until it's off all of the time.
I may have already left the dome on too long. Some (not all) of my sprouted seedlings have what looks like a fine white fuzz on their stems, near the soil. Is this the fungus you mentioned? Otherwise they look fine. What do seedlings that had too much humidity look like, and is the fungus contagious? (In other words do I have to dump the whole tray and start over again or can I wait and see if the other seedlings have develop those symptoms too?)
I am surprised at how WET my my seedling trays are this year (I don't remember them as such when I've sowed seeds before). I know I drained the trays well after pellets explanded.