Other mistake: Wolverine shields Yashida from the blast, burning off all his hair. Somehow his pants manage to remain unaffected by this, as seen later when they are climbing from the pit.
Other mistake: When Wolverine has the parasite inside him he can't heal, but the holes for the claws heals just fine.
Other mistake: When Logan loses his ability to heal, all his wounds stay, but his hands still heal after every usage of his claws - the skin on his hands is untouched.
Answer: The healing factor was only reduced. Like you said, had it been removed he would have bled to death and every time he used his claws he would be bleeding from his hands. It is not revealed if it is at full strength at the end of the film, but it is likely it will return to full strength over time.