HICKORY, N.C.—In the late 1990s, this furniture-making hub seemed sheltered from the disruptive forces of globalization. Laid-off steelworkers from West Virginia, Tennessee and beyond streamed here for new jobs building beds, tables and chairs for American homes. The unemployment rate fell below 2%.

These days, Hickory is still suffering from a series of economic shocks, none more powerful than China’s rise as an export power. The invasion of imported furniture drove factories out of business, erased thousands of jobs and...