I have spent the last week going in circles trying to find this story. I can only remember one scene clearly, Bucky goes back to a Hydra site (the bank I think) and finds the chair. He picks up the rubber mouthguard and cuts the handle(?) off it so that he can close his lips around it. He then destroys the chair, and leaves. He chews/holds the mouthguard in his mouth for comfort/grounding.
I think he was searching down what was left of Hydra to destory them. Sam and Steve were following him/ also trying to destroy Hydra... (I'm not sure which)
It would have been on AO3.