She beat former Congressman Lou Frey by 62-38% in a runoff. It helped that the Dems had an ugly fight that saw the incumbent Senator, Richard Stone, lose to his 1974 primary opponent in a rematch (Bill Gunter). Gunter outspent Hawkins by almost 4 to 1 in the general and still lost 52-48%.
Curiously, ex-Sen. Stone was later rewarded with Ambassadorships by both President Reagan and GHW Bush.
Apparently former FL Senator Stone was somewhat of an independent thinker and not always a part-line follower.