About 18 months ago, Catskill Animal Sanctuary in Saugerties, N.Y., rescued 42 neglected and ailing sheep. Many were anemic and had foot rot, a contagious bacterial disease that can be life-threatening if left untreated in wet environments.

For the animals to recover, they should be in a clean and dry place, said Kathy Stevens, the 150-acre sanctuary’s founder and executive director. But the sheep and their new caretakers faced a rapidly developing problem: soggy pastures and flooding barn stalls.

Increased rainfall, among the weather distortions caused by climate change, has finally forced the sanctuary to search for a new home, a predicament shared by a growing number of animal refuges across the United States.

“It is absolutely untenable to stay here and to wait until a true disaster strikes,” said Ms. Stevens, who fears that the regular flooding the sanctuary has been experiencing could soon become dangerous.

Climate change has resulted in warmer and wetter weather across New York, where annual precipitation jumped 10 to 20 percent over the past century, according to a state report. The study projects that the largest precipitation increases in coming years will be in New York City, the Catskill Mountains region and the Lower Hudson Valley.