Queensland are 6-221 in their second innings, a lead of 45 runs, at lunch on Friday's third day of play in Adelaide.
Doggett claimed six wickets in the Bulls' meagre first innings of 95 and SA, with Jake Lehmann scoring a century, replied with 271.
Doggett has followed his first innings feats with four more wickets in the second dig, the first time the 30-year-old has captured 10 wickets in a first-class game.
The former Queenslander is just the third bowler to take 10 wickets in a Shield final, following Tasmania's Shane Jurgenson's 11 wickets in the 2001/02 final and Queensland's Mitchell Johnson's 10 victims in the 2005/06 decider.
Doggett has 4-53 after adding two more victims on Friday after the Bulls resumed at 3-177.
The right-arm quick took just 15 minutes to break through, producing an off-cutter which beat the defensive bat of Ben McDermott (17) and trapped the Queenslander plumb lbw.
In his next over, Doggett repeated the feat against Jimmy Peirson (five) who, prodding forward defensively, was defeated by a ball that nipped back and hit him on the pads.
Queensland, despite Jack Clayton's unbeaten 57, hit another hurdle on the last ball before lunch.
SA took the second new ball and Nathan McAndrew (1-57) removed Michael Neser (13) when the Queenslander flashed at a widish delivery and edged to wicketkeeper Alex Carey.
SA are seeking to win the Shield for the first time in 29 years.
The South Australians are also aiming to collect the Shield and the one-day trophy in the same season for the first time after their victory in the 50-over final against Victoria on March 1.
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