ALBANY, N.Y. — Republicans in New York probably had hoped the worst was over.

In 2018, the party suffered its most devastating electoral cycle in recent memory, as Democrats rode a fervent anti-Trump sentiment to capture eight seats — and the majority — in the State Senate. They also won every statewide office, including a third term for Gov. Andrew M. Cuomo.

This year, Republicans signaled their willingness to change — electing a young new state party chairman and promising to thwart “the corrupt, socialist agenda permeating the Democrat Party in New York.”

That fight is off to a rough start.

On Wednesday, State Senator Joseph E. Robach, who represents a district in the Rochester area, became the fourth Republican senator in two weeks to announce plans to not seek re-election in 2020 — joining a rash of other Republican senators who have left the chamber or decided to run for other public offices.

The losses of Mr. Robach, 61, and another longtime state senator, Elizabeth Little, 79, are particularly acute for the Republicans because of their extended tenure; combined, the two have served more than 50 years in the State Legislature.