Will my cousin and I be sinful for accusing someone of committing a grave sin due to a misunderstanding, and how can we atone for this mistake?

If you accuse the man of that, you have harmed him, for Allah Almighty says: "And those who harm believing men and believing women for [something] other than what they have earned have certainly borne slander and manifest sin" (Al-Ahzab: 58). The interpretation of this is attributing to them what they did not do. So you both must offer sincere repentance.