Adcox's father, Herb Adcox, owned a Chevrolet dealership in Chattanooga and was his son's first car owner and sponsor. Grant's Winston Cup career began in 1974 with a handful of races backed by the family business. At that year's Winston 500 at Talladega Superspeedway, an accident on pit road during a caution period sent Adcox's car into Gary Bettenhausen's stationary Matador, crushing catch-can man Don Miller between the vehicles; Miller subsequently lost his right leg. Adcox went into shock upon learning the extent of Miller's injuries and his car was withdrawn from the event.
He qualified for the 1975 Talladega 500 but was forced to withdraw after his crew chief Gene Lovell suffered a fatal heart attack in their garage. Adcox found a substitute ride, but the race was delayed a week by rain and he was again forced out. His grid slot passed to Tiny Lund, who died during the race on lap seven.
Adcox's best Winston Cup result was a fifth-place finish in the 1978 Winston 500 at Talladega, his career highlight at the top level. An underfunded operation limited his appearances throughout his career, and he never contended seriously for a Winston Cup win.
Adcox's primary competitive success came in the ARCA series, where he was particularly formidable at superspeedways. He won a record four superspeedway events in 1986 alone and accumulated eight superspeedway victories in total during his ARCA career, five of them at Talladega. Dale Earnhardt, speaking before a 1989 race at North Carolina Motor Speedway, stated that if Adcox had received the monetary backing available to better-resourced teams, he might have become a success in NASCAR.
For 1989 Adcox had signed with Bumper to Bumper All Pro Autoparts and planned to run a handful of Winston Cup events in a family-owned Chevrolet.
At the 1989 Atlanta Journal 500 on November 19, Adcox crashed heavily on lap 202 of the race and died of major chest and head injuries, compounded by a heart attack suffered as a result of the impact. Investigation revealed that the severity of his injuries stemmed directly from his racing seat having been improperly mounted and torn away from its mount entirely in the impact. The finding led NASCAR to introduce new regulations governing seat mounting for the 1990 season.
Adcox is interred at the Chattanooga Memorial Park in Chattanooga, Tennessee.
ARCA's sportsmanship award, the H.G. Adcox Award, was established in Adcox's honor and was presented annually by his father Herb until Herb's death in 2015. The Grant Adcox Memorial race is held annually at North Georgia Speedway, with a parallel event also staged in Cleveland, Tennessee, in his memory. His death at Atlanta, and the safety findings that followed, directly influenced NASCAR's approach to seat mounting standards across the sport.