Answer: I'm inclined to think the rain has washed the seeds down to the soil level, which is a good thing. If they have good contact with the soil, they will root well when the weather warms up enough for them to germinate. Mowing shouldn't have much effect on the seeds, if in fact they have washed down to the soil rather than remained up near the tops of the grass plants. It's a little early to broadcast grass seeds; the middle of April is about as early as you'll want to do so. Grass will sprout and grow when temperatures are about 40F and they generally germinate within about a week. So, go ahead and mow. If you don't see any new grass by mid-April, you'll want to reseed in mid to late April.
Best wishes with your lawn!
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