I raise sunflowers for animal
feed, but I do not have goats

Mostly Black Oil Sunflowers.
Once the seeds are fulling formed, darken in color, you can pull up the plant and just sit it somewhere to dry. I keep mine out of the sun and rain. once dry
chickens and ducks can remove the seeds out of the heads themselves, or you can pop them out and save for next years planting.
I would
think goats would prefer you feed them the stocks at maturity when they are still greenish. So cut your SF heads off for drying and toss the green stocks to your goats.
When they loose their petals they are drying out, and well past maturity, you don't have to wait that long for harvesting the entire plant.