Ya, give her a chance and see how broody she does end up being before deciding to re-home if needed. I can work with once or twice per year but have had to re-home as some would go back to back to back every 3rd egg even after being allowed to raise a brood.
If she hasn't taken in enough feed since becoming broody or her system just isn't adjusting yet.. maybe why the droppings are seeming off. But
NO, I don't think I would call that normal. Are you seeing her eat or drink? What is her feed including treats and supplements? Have you felt her crop, and is it full, empty, hard, soft, squishy, any sour smell?
I would clean her up and dry her off. I usually use warm running water, both my sink and tub have spray heads on hoses. A small soaking tub may also work if it's hardened on too much a little extra time helps or dish soap, rinsed
well. Just towel dry if temperatures are comfortable or gently blow dry if it's colder. They are usually not very good at preening it out. But often relax well enough for the bath. I'm against cold water dunks to break broody's.. I always use warm (not hot) water to reduce stress during bathing.
One word regarding broodiness and egg production.. my excessively broody Silkie ladies also actually mini molted during raising their broods, returning to lay in the usual timing.. Never having full winter molts.. essentially POSSIBLY negating any differences verses a bird that spends a full 3-4 months molting each year. Plus the bonus of having them raise a few new pullets added each year to keep up winter production vs heat lamp etc. Gosh, even though I've been done with Silkies for a while.. Sometimes I wonder why.. alas it was the broodiness that broke me. I don't like breaking them but couldn't accommodate it either.
Since switching to Flock Raiser (Purina 20% protein) with oyster shell on the side free choice instead of using "layer" (usually 16% protein with OS included).. My older birds now enter molt less harshly, returning to lay sooner and having better over all condition.. Plus the extra nutrients actually DO go into the eggs my family is consuming.. according to studies I've seen on hatch rates, etc. For informational purposes only.
Unwashed, refrigerated eggs stored pointy side down are still viable even 6 months later, and still as fresh as most store bought eggs! The air cell grows larger, leaving a flat spot when boiled. The yolk and white might not stand AS tall as they did on day one.. But planning ahead can be helpful.. She says as she's just bought 2 dozen market eggs, it doesn't always work out!
Let us know how she cleans up and the food/crop status when ya can.