The problem is with slashes: your variable contains them and the final command will be something like sed "s/string/path/to/something/g", containing way too many slashes.
Since sed can take any char as delimiter (without having to declare the new delimiter), you can try using another one that doesn't appear in your replacement string:
replacement="/my/path"
sed --expression "s@pattern@$replacement@"
Note that this is not bullet proof: if the replacement string later contains @ it will break for the same reason, and any backslash sequences like \1 will still be interpreted according to sed rules. Using | as a delimiter is also a nice option as it is similar in readability to /.
