Dean and Claire left the cave after roughly one day of rest. They weren't completely rested, but the real problem there was that they didn't have enough food or water to stay inside that place forever. Also, it was unlikely that they would find something to eat in the mountains. In order to survive, they needed to go outside to find something to eat.
As they were forced to fight against the blizzard again, their tiredness just increased each second more. They felt like they were getting to their very limits, particularly Dean. Now that Excalibur was gone, he didn't have any way to enhance his body instead of using "Chaos" to do so. Obviously, as shown in the past, using "Chaos" in any way was very dangerous as he didn't have total control over it. That power was like an infectious virus that drove him crazy every time he used it.
