This sounds like it could be Stephen Baxter's Titan.
Possible signs of organic life have been found on Titan, Saturn's largest moon. A group of visionaries led by NASA's Paula Benacerraf plan a daring one-way mission that will cost them everything. Taking nearly a decade, the billion-mile voyage includes a "slingshot" transit of Venus, a catastrophic solar storm, and a constant struggle to keep the ship and crew functioning. But it is on the icy surface of Titan itself that the true adventure begins. In the orange methane slush the astronauts will discover the secret of life's origins and reach for a human destiny beyond their wildest dreams.
There are some significant differences from the Question though. The initial mission is launched in the name of science rather than to escape an apocalypse. Though it's one way nature lends itself to confusion on the issue.
Additionally the rise of an anti-science government in the US and possibly more widely means that launching the mission has a lot of "last chance" flavour. Which again is a common theme for escapes from the apocalypse.
I'm pretty sure they start out with a crew of four or five. This is reduced through accident and infighting a fairly horrific attempted rape forming part of this drama. (This would be the one man going a bit wild).
The destination is Saturn's moon Titan.
As a small nugget that might prompt recall, one crew member eventually dies of vitamin A poisoning after eating too many carrots that were generically modified to produce more vitamin A. He recalls this happening to an Antarctic expedition during the early days of polar exploration from eating dogs livers.
The book ends with is distant finale where the crew members who actually reached Titan are resurrected by the sufficiently advanced future race that evolve there following the sun's expansion.