$ DOCS="/cygdrive/c/Users/my\ dir/Documents"
Here's your first problem. This puts an actual backslash character into $DOCS, as you can see by running this command:
$ echo "$DOCS"
/cygdrive/c/Users/my\ `
When defining DOCS, you do need to escape the space character. You can quote the string (using either single or double quotes) or you can escape just the space character with a backslash. You can't do both. (On most Unix-like systems, you can have a backslash in a file or directory name, though it's not a good idea. On Cygwin or Windows, \ is a directory delimiter. But I'm going to assume the actual name of the directory is my dir, not my\ dir.)
$ cd $DOCS
This passes two arguments to cd. The first is cygdrive/c/Users/my\, and the second is dir/Documents. It happens that cd quietly ignores all but its first argument, which explains the error message:
-bash: cd: /cygdrive/c/Users/my\: No such file or directory
To set $DOCS to the name of your Documents directory, do any one of these:
$ DOCS="/cygdrive/c/Users/my dir/Documents"
$ DOCS='/cygdrive/c/Users/my dir/Documents'
$ DOCS=/cygdrive/c/Users/my\ dir/Documents
Once you've done that, to change to your Documents directory, enclose the variable reference in double quotes (that's a good idea for any variable reference in bash, unless you're sure the value doesn't have any funny characters):
$ cd "$DOCS"
You might also consider giving that directory a name without any spaces in it -- though that can be hard to do in general on Windows.