WASHINGTON—On the first day back in the Senate after Sen. Lindsey Graham’s (R-S.C.) sudden and unexpected passing, legislative business continued—but the atmosphere was somber. “I just walked past Lindsey’s desk,” Sen. Ron Johnson (R-Wis.), in line to inherit his Budget Committee chairmanship, told reporters on July 13. “I’ve gotta think one step at a time.” Graham, 71, was known to many colleagues as a tireless dealmaker, emblematic of an older Washington where it was easier to work across the aisle. In recent years, the four-term senator and foreign policy hawk who ran against President Donald Trump in 2016 emerged as a key conduit between the president and an upper chamber that is sometimes at odds with him. Recent tensions have centered on the SAVE America Act, which aims to bolster election integrity....