Merge 2 dfs with several IDs (using a lookup or conditions?)

I want to do a right join with these 2 dfs:

df1 <- structure(list(ID_1 = c("NE1", "NE2", "LR2", "NE4", "WE5"), ID_2 = c("WW1", 
                                                                            "NE3", "LR2", "NE5", "WE5"), Country = c("Togo", "USA", "Iceland", 
                                                                                                                     "Chad", "UK")), class = "data.frame", row.names = c(NA, -5L))


df2 <- structure(list(ID_1 = c("NE1", "NE3", "LR2", "NE6"), Sex = c("M", 
                                                                    "F", "M", "M")), class = "data.frame", row.names = c(NA, -4L))

Note df1 has 2 ID columns, whereas df2 has only 1 ID column.

I want the match to be made on ID_1, and then, only if the match cannot be made on ID_1, to be made on ID_2. In other words, I cannot use c(ID_1, ID_2) or similar.

Desired output:

#     ID_1    Country   Sex
# 1   NE1     Togo      M     --> based on ID_1 of df1
# 2   NE3     USA       F     --> based on ID_2 of df1
# 3   LR2     Iceland   M     --> based on both ID_1 and ID_2 of df1
# 4   NE6     <NA>      M     --> no match but row is kept (right join)

I also want to end up with a single ID column as in the desired output above.

I found solutions for Python (https://stackoverflow.com/questions/58739744/how-to-merge-2-df-based-on-comparison-of-2-columns-to-match-1-column and https://stackoverflow.com/questions/48585027/merge-pandas-df-based-on-2-keys) but not R, or cannot make them work.

library(dplyr)

df2 %>%  
  left_join(distinct_all(select(pivot_longer(df1, -Country, values_to = "ID_1"), -name)))

Joining with `by = join_by(ID_1)`
# A tibble: 4 × 3
  ID_1  Sex   Country
  <chr> <chr> <chr>  
1 NE1   M     Togo   
2 NE3   F     USA    
3 LR2   M     Iceland
4 NE6   M     NA     

More readable format:

df1 <- df1 %>% 
  pivot_longer(-Country, values_to = "ID_1") %>% 
  select(-name) %>%  
  distinct_all()

df2 %>% 
  left_join(df1)