pivot_wider command generates a list() instead of a "wider" data.frame in R
Add an id column and things get easier:
df %>%
group_by(season) %>%
mutate(id = row_number()) %>%
pivot_wider(names_from = season, values_from = corr, id_cols = id)
# # A tibble: 28 × 3
# id wet dry
# <int> <dbl> <dbl>
# 1 1 0.747 0.747
# 2 2 0.747 0.747
# 3 3 -0.747 -0.747
# 4 4 0.711 0.711
# 5 5 -0.673 -0.673
# 6 6 -0.627 -0.627
# 7 7 0.721 0.721
# 8 8 0.747 -0.570
# 9 9 0.747 -0.722
# 10 10 -0.747 0.609
# # … with 18 more rows