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

Copyright Notice：Content Author:「Troels」，Reproduced under the CC 4.0 BY-SA copyright license with a link to the original source and this disclaimer.
Link to original article：https://stackoverflow.com/questions/72047691/create-column-based-on-specific-values-in-other-columns