Two years ago, Austin real-estate agent Amy Deane, of Moreland Properties, was working with so many wealthy out-of-state buyers that she showed one $15 million house five times in 30 days. Now, she might get one call every other week for showings in that price range. “That big buyer pool has slowed down,” she said. “The first movers committed and moved.” After an epic two-year run—not just in Austin but in major cities around the country—the luxury real-estate market is finally cooling. Real-estate agents in places like New York, Los Angeles, and the Hamptons say the frenzied deal making and record-setting prices that characterized the past few years has eased,…