Sorry, not enough sheepie experience to help very much. Mine are on very lush grass at the moment and no serious problems yet

. They dont seem to keep the grass very short even though I think my stocking is fairly high - 7 ewes and 9 lambs on just under 2 acres of hill type grazing. We do have thick hedges and I guess they get a lot of their nutrition from these as they seem to prefer browsing to grazing. Someone with experience of keeping Soay told me that the land would support double the number of ewes that I have.
Maybe strip grazing would help if you are having problems.

Would they eat hay to dry them up a bit

.
Someone else, with more experience, may come along and be able to give you some better advice or you could start a new thread about this.