CARSON CITY, Nev. (AP) — The pressure is on Republican senators — from congressional leaders, conservative groups and impatient GOP voters — to fulfill a seven-year-old promise to scrap much of Democrat Barack Obama's health care law. But back home, Republican governors who have experienced some of the upside of the law are warning their GOP senators to first, do no harm.

For these governors, the issue is less about delivering a triumph to President Donald Trump and more about not blowing a hole in state budgets and maintaining health care coverage for constituents. In the critical next few we...