The furious efforts in Georgia by the Sunrise Movement and other progressive groups — on behalf of two candidates who do not share their most ambitious policy goals — reflect the urgency that is consuming the Democratic Party’s left flank. Two victories in Georgia would produce a 50-50 tie in the Senate, giving Democrats control of the chamber because Kamala Harris would cast tiebreaking votes as vice president.

Without Democratic control, progressive lawmakers, activists and their grass-roots supporters worry that they will not be able to achieve even a pared-down version of their policy wish list for the country.

But they also understand that for decades Georgia has been a Republican stronghold with a large number of conservative voters, and that their efforts there need to be modulated. President-elect Joseph R. Biden Jr. won the state, many Democrats point out, with a moderate agenda that tempered the rhetoric and policy goals of the left. Mr. Biden, Mr. Warnock and Mr. Ossoff do not support “Medicare for all,” another priority of the party’s left wing.