It’s Gordon R. Dickson’s novelette “In the Bone”.

• first published in Analog Science Fact & Fiction, September 1963 (John W. Campbell, ed.)
• subsequently anthologised many times – e.g. Analog 4 (1966), The Best of Gordon R. Dickson (1978), The Analog Anthology #2 (1981) – so it would have been easy to encounter it in an ­-80s paperback collection.

Story outline (matching the details in the question)

– A small, all-male survey ship is trying to get back to Earth, but a navigation accident has left them with only a chain of pre-programmed jumps from star-system to star-system.
– Between jumps the men remain in cold-sleep; at each stop the whole crew is awakened and two of them (usually the captain plus one other) are sent down with a lander to see whether they have finally reached Earth.
– Again and again they are disappointed. One or two crewmembers commit suicide after almost every failure, and morale is disintegrating.
– On the story’s final planet the captain and his companion walk through the radioactive, dust-choked ruins of a city. Only gradually do they recognise familiar outlines: they have, in fact, reached Earth – long since destroyed by the war everyone feared.
– The junior man laughs hysterically and shoots himself. The captain returns to orbit, puts the rest of the crew back into the freezer, programs the next automatic jump, and records a cheerful wake-up message urging the survivors to “keep your chins up, maybe the next one will be home,” because the truth would finish them.

Everything in the question—the cold-sleep routine, the two-man landing parties, the cumulative suicides, the thick dust over the ruins, the final suicide and the captain’s decision to keep the discovery secret—comes straight from “In the Bone.”