I'm pretty sure you would go into shock, be unable to do anything except fall to the ground and quickly die of blood loss
For humans, I think that you're about right, certainly, someone whose arm got severed probably wouldn't continue fighting, and would most likely die from blood loss without receiving quick medical attention, still, I could see this being reflected in a special death animation where instead of just instantly dropping dead, enemies who died from severed limbs could fall to the ground, move around a bit while bleeding from the wound, and then go into limp ragdoll mode. We still wouldn't get people with severed arms running around shooting at you, but it would at least help to avoid the issue common in games where any lethal wound causes you to instantly drop dead, even if it's the kind that in real life really would result in a slow death. Also, while a severed limb might not just be a flesh wound for a human, I could see some types of enemies being able to continue fighting even after losing one or more limbs, after all, not all enemies are structured the same way as humans, and the same injuries wouldn't always effect them in the same way.
So I'd say it's not necessarilty a bad idea, but I certainly know it won't happen in New Vegas, it could in the future game, though.