WASHINGTON—After more than 20 hours of continuous wrangling, Congressional Democrats and White House officials reached agreement on the final shape of legislation that would transform financial regulation, avoiding last-minute defections among New York lawmakers that had threatened to upend the bill. After months of uncertainty about how the U.S. would craft new rules, the agreement offers the clearest picture since the financial crisis of how markets and the government will interact for decades to come. The common thread: large financial companies are facing a tougher leash.