After attending Pensacola Junior College, Broxson transferred to Evangel University in Springfield, Missouri, where he met his wife, Mary, and graduated in 1971.

In the general election, Broxson won a landslide victory against independent candidate Matthias Venditto, receiving 84% of the vote.

While serving in the legislature, Broxson supported legislation that would allow communities "to fine motorists who play their car stereos too loudly" despite concerns that it would lead to racial discrimination, noting, "This has been an existing law for a long time.

"[4] He ended up defeating Smith in a landslide, winning 70% of the vote, and advanced to the general election, where he only faced write-in opposition and won re-election easily.

Broxson defeated fellow state representative Mike Hill in the Republican primary and two write-in candidates in the general election.