He was a leading advocate of lowering the U.S. voting age and wrote the [[26th Amendment]].<ref name=nytimes/>
Randolph faced a strong challenge in his 1978 re-election campaign from Governor Arch A. Moore.<ref>Roberts, Sam (January 8, 2015). [https://www.nytimes.com/2015/01/09/us/arch-moore-91-w-virginia-trail-blazer-dies.html Arch Moore, Trailblazing West Virginia Governor, Dies at 91]. ''The New York Times''. Retrieved August 2, 2021.</ref> Although he heavily outspent Moore, Randolph only won the general election by one percentage point.<ref>[https://www.ourcampaigns.com/RaceDetail.html?RaceID=5331 WV US Senate Race - Nov 07, 1978]. ''Our Campaigns''. Retrieved August 2, 2021.</ref>
In 1983, Randolph voted for the [[Orrin Hatch|Hatch]]-Eagleton [[Human Life Amendment]] which would have overturned ''[[Roe v. Wade]]''.<ref>[https://www.govtrack.us/congress/votes/98-1983/s169 TO PASS S. J. RES. 3, A MEASURE AMENDING THE CONSTITUTION TO ESTABLISH LEGISLATIVE AUTHORITY IN CONGRESS AND THE STATES WITH RESPECT TO ABORTION, THEREBY OVERTURNING THE SUPREME COURT'S DECISION IN ROE VS. WADE. (MOTION FAILED;2/3 REQUIR.]. ''GovTrack.us''. Retrieved August 2, 2021.</ref> However, it only received a total of forty-nine affirmative votes, falling far short of the two-thirds majority required for congressional approval of constitutional amendments.