[sh] How can I have a newline in a string in sh?

On my system (Ubuntu 17.10) your example just works as desired, both when typed from the command line (into sh) and when executed as a sh script:

[bash]§ sh
$ STR="Hello\nWorld"
$ echo $STR
Hello
World
$ exit
[bash]§ echo "STR=\"Hello\nWorld\"
> echo \$STR" > test-str.sh
[bash]§ cat test-str.sh 
STR="Hello\nWorld"
echo $STR
[bash]§ sh test-str.sh 
Hello
World

I guess this answers your question: it just works. (I have not tried to figure out details such as at what moment exactly the substitution of the newline character for \n happens in sh).

However, i noticed that this same script would behave differently when executed with bash and would print out Hello\nWorld instead:

[bash]§ bash test-str.sh
Hello\nWorld

I've managed to get the desired output with bash as follows:

[bash]§ STR="Hello
> World"
[bash]§ echo "$STR"

Note the double quotes around $STR. This behaves identically if saved and run as a bash script.

The following also gives the desired output:

[bash]§ echo "Hello
> World"