Create column based on specific values in other columns

I am trying to create a new column based on specific values in several (four) other columns.

Reprex:

col1 <- c("a", "", "a")
col2 <- c("", "b", "b")
df <- data.frame(col1, col2)

I have tried with case_when, but cannot get it to apply to all values.

df$col3 <- with(df, case_when(col1 %in% c("a") & col2 %in% c("b") ~ "c" ))

What I want to achieve:
| col1|   col2   |  col3   |
|:--: | :-------:|:-------:|
|   a |          |    a    |
|     |     b    |    b    |
|   a |     b    |    c    |

So if col1 is a, col2 is na then col 3 is a.
If col1 is na, col2 b then col3 should be b and finally if col1 is a, col2 b then col3 should be c

>Solution :

Here is how we could do it with case_when, first just replace blank cells with NA

library(dplyr) 

df %>% 
  mutate(across(starts_with("col"), na_if,""),
         col3 = case_when(
           col1=="a" & is.na(col2) ~ "a",
           is.na(col1) & col2 == "b" ~ "b",
           col1 == "a" & col2 == "b" ~ "c",
           TRUE ~ NA_character_))

  col1 col2 col3
1    a <NA>    a
2 <NA>    b    b
3    a    b    c