Weapon is not in character's hand while in battle stance?

Try un-equipping your weapon and then re-equipping it from your inventory. This should fix the animation.

I have often managed to trigger this bug, or similar bugs, when experimenting with my own (not publicly-accessible) animation modding.

PorkChopPenfold's suggestion to unequip and re-equip your weapon is a good idea, and usually works in my experience. There is a more general technique that can fix almost any animation bug, but only on PC. Open the console and run this command:

pushactoraway player 1

Then close the console and wait for the player to finish the "stand up" animation (it takes a few seconds). In the vast majority of situations, this will completely fix the problem, because it puts the player into a ragdoll state and cancels any ongoing animations. However, you should consider the following caveats:

• Always save the game before using the console.
• If you are swimming (or in a glitched running-underwater state), get to dry land first. This technique will not fix swim states, and may actually break them.
• I have not extensively tested how this interacts with mounts. If you are riding a horse or dragon, this might not work as expected.
• If your character is in an animation-driven state (i.e. you can't move around normally because the dragonborn is "doing something" like using a forge or sitting on a chair), then it is probably safer to get to a movement-driven state first. In other words, close the crafting menu, stand up from the chair, etc., so that your character moves normally when you hit the WASD keys (or manipulate the joystick on your gamepad).
• If you use mods which trigger custom animations, arm offsets, etc., then this might cause issues with those mods if they do not correctly handle the "get up" animation. That would be a bug in the mod, because Unrelenting Force (which can be used against the player by certain draugr) can also trigger the ragdoll state, so those mods should already be handling it correctly anyway.

Non-PC players can probably get away with saving and reloading the game, but you might need to completely exit the game (or restart the console) as well, because animation state might persist over a save/load.

You can replace "player" with the refID of a misbehaving NPC, if you want to fix their animations instead. Unfortunately, clicking on NPCs to target them does not work for the purposes of this command (instead, it sets the point from which the Havok impulse originates, which is largely useless as we just want to trigger the ragdoll state).