Could Judas have sought forgiveness instead of despairing?

Of course he could. In his Life of Christ, Bishop Fulton Sheen devotes a chapter to Judas's betrayal, comparing it with Peter's, which was similar in some ways. He concludes that the tragedy of Judas was that even after the betrayal he could have become St. Judas Iscariot, had he repented instead of despairing.

The sin against the Holy Spirit, which cannot be forgiven, is final impenitence. God can forgive any sin that is repented, but a person must repent to be forgiven. Only God judges hearts, so we cannot know whether Judas remained impenitent until death. But we know he did not show the heroic virtue of Peter, who repented of his betrayal rather than let despair consume him.