Astronaut discovers debris of an alien ship in space and ends up finding Bigfoot kind of describes Benford's In the Ocean of Night. It was originally published in 1977, but widely reprinted in the mid-to-late 1980s.
I'm afraid this, like most of Benford's work, left me cold, so I'll just quote the plot summary on Wikipedia:
In 1999 (2019 in the second edition), Nigel Walmsley, a British scientist and astronaut for NASA, is sent to attach a thermonuclear bomb to a comet named Icarus which is on a direct collision course for India. Icarus turns out to be large, solid, and made of a nickel-iron composite. Nigel is instructed to plant the weapon and leave so it can be detonated. He persuades Mission Control to let him put it in a large fissure he discovered, so it would be even more effective.
In the fissure, Nigel discovers strips of metal worked in obviously artificial patterns. Awestruck at this evidence of extraterrestrial intelligent life, Nigel begins exploring. Icarus is made up of a number of hollow shells, making the asteroid's mass far less than predicted. NASA insists that the demolition has to go forward, claiming Icarus would skip off the atmosphere and land in the Indian Ocean, causing widespread damage from the resultant tsunami. Nigel realizes this is a lie, and convinces his partner of that. They hide the nuke and spend the next week retrieving artifacts and materials before detonating the bomb.
[...]
Nigel meets the Snark, which disables Nigel's conventional weapons and begins talking to him. It says that organic civilizations and species are inherently unstable; they flash brilliantly and commit suicide sooner or later. The autonomous machines they craft live on long after them, going on and evolving. But they cannot truly compete with the organics, who live "in the universe of essences". That is the reason for the Great Silence. Nigel's superiors order him to use the nuclear weapon, but he refuses. They override him and fire it anyway but the Snark flees the Solar System faster than the missile can follow. The decision to fire is covered up, but Nigel blackmails NASA into letting him go to the Moon; the Snark had directed a transmission at Mare Marginis for unknown parties, and Nigel wanted to find those parties.
Four years later, Nigel is now based on the Moon. A fellow astronaut, Nikka, is involved in a crash that accidentally discovers a still active alien spacecraft wreck in the Moon's Mare Marginis – a spacecraft suspiciously armed with a once-powerful anti-spacecraft weapon. Nigel and Nikka explore the wreck, and find a functioning computer with a direct neural interface. Nigel and several others experiment with the computer's neural hook-up, and leave fundamentally changed by it, while the computer becomes inert and unable to reveal any more about its creators. Meanwhile, on Earth, some surprising experiments in human genetics conducted by the aliens are discovered alive in North America.