Donoghue represents Subdistrict 2C, which roughly matches the borders of the city of Hagerstown.
He defeated Muldowney in a 1998 Democratic primary. Two years later, Muldowney switched his enrollment to Republican.
Although Hesse halted his campaign, his name will be on the primary ballot. The deadline for candidates to officially withdraw was July 13.
Hesse made his announcement Tuesday afternoon at the University System of Maryland at Hagerstown campus on West Washington Street in front of about 40 supporters and observers.
Muldowney joined him. He said Hesse first mentioned the idea to him Saturday, but "it's one of those things you don't believe until you see it."
Monday afternoon, however, Muldowney had said he heard about the press conference, but didn't know what the announcement would be.
"What he did today is huge," Muldowney, who served two terms as a delegate from 1979 to 1987, said Tuesday of Hesse. Muldowney said he can save his campaign funds and his energy for taking on Donoghue.
The general election is Nov. 7.
The Washington County Republican Central Committee didn't pressure Hesse to pull out of the race, Chairwoman Paula Lampton said.
"Absolutely, this was a decision by Scott Hesse alone," she said.
Bob Sweeney, the central committee's first vice chair, read aloud a message he said Republican Gov. Robert Ehrlich sent Monday night thanking Hesse "for his decision to work with the Republican Party in Washington County."
Muldowney filed to run March 6, but raised money before then.
As of Jan. 18, the most recent campaign finance filing deadline, the Friends of Paul Muldowney account had a balance of $28,351.38.
Friends of Scott Hesse had a $100 balance, plus outstanding obligations of $1,187.82. However, Hesse said Tuesday that since then, his campaign raised all of the money it would need. He didn't know how much that was.
Hesse said he waited until April 21 - his 51st birthday - to file, even though he had made up his mind earlier.