The Tigers (6-4, 6-2 Southwestern Athletic Conference) moved ahead of the Bulldogs (7-3, 6-3) in the Eastern Division race and hold a tiebreaker advantage over Alabama State if they can beat Alcorn State next week.
Moore also ran for a 7-yard touchdown to open the scoring in the second quarter, and Sims's 7-yard scoring run made it 14-0 at halftime.
Alabama A&M halved its deficit in the third quarter, but Jackson State regained control with two big plays.
Bulldogs quarterback Deaunte Mason threw for three touchdowns, but the Tigers limited him to 17 completions in 40 attempts for 193 yards.
South Carolina coach Steve Spurrier told his players and coaches Monday night that he is retiring immediately, sources told ESPN.com.
Steve Sarkisian has been plagued of late by apparent substance-related circumstances, including arriving to team facilities appearing intoxicated Sunday, according to sources.
Florida Gators quarterback Will Grier has been suspended for the year after testing positive for a performance-enhancing substance found in a supplement, coach Jim McElwain announced Monday.
Former USC coach Pete Carroll said Monday that he's saddened by Steve Sarkisian's situation and he has reached out to his former assistant.
Ole Miss offensive tackle Laremy Tunsil will be eligible to return to action for the Rebels following Saturday's game against Memphis, sources told ESPN.com on Monday.
After a decades-long ban, Nevada Gaming Control granted approval for the state's regulated sportsbooks to offer wagering on the Heisman Trophy, with LSU's Leonard Fournette the early favorite.