Given that you have:
line="Some Text that should be changed \\"
so your echo "$line" yields:
Some Text that should be changed \
Then sed sees:
sed -i "s/Some Text that should be changed \/Your replacement/" yourfile
and the backslash means that your search pattern hasn't ended, and therefore the substitute command is malformed - as the error message says.
You will have to get a second backslash onto the end of the string. One of a myriad ways of doing that is:
case "$line" in
(*\\) line="$line\\";;
esac
This is just being inventive...but it has the merit of not executing any external command to fix the string. There are also more direct substitutions available in bash.
Now you can do:
sed -i "s/^$line$/$newline/" $myFile
and as long as $newline contains neither any slashes nor any backslashes, you will be safe enough.