Republicans denounced the proposed overhaul as an attempt to mount a partisan takeover of what critics consider a moribund Federal Election Commission.
Races in Missouri and Alabama, with others to come, reflect the potential risks for a party in which loyalty to Donald Trump is the main criterion for securing nominations.
The hard-right Republicans’ entry to the races for open Senate seats heralded fiercely contested G.O.P. primaries in the two deeply conservative states.
The disputes are reminiscent of the fight surrounding the creation of the independent commission that conducted an inquiry into the Sept. 11, 2001, attacks.
Senator Richard C. Shelby, Alabama’s longest-serving senator, will retire after a career in which he headed four congressional committees and directed billions of dollars to his home state.