Logan is dying from Adamantium Poisoning. Prior to the scene in question though, he was brutally beaten by a younger clone of himself, adding to his existing injuries. He collapsed a day later, and it took injections of a green healing serum to help get him better. It still took 3 days for it to work enough to wake him up. They were carefully injecting him with a little at a time due to the side effects of the drug (extreme rage, loss of control, and adverse effects on normal healing). By using it, it helped heal but left him weak, exhausted.
And then he used a full vial of it, pretty much guaranteed he would be near death when the body in strength ran out. Red Bull sucks like that.
So his limp running is a direct result of a week of constant injuries and drugs on top of his illness.