If $N$ is divisible by both 5 and 11, then $N$ is divisible by $5 \times 11=55$. This is because 5 and 11 have no common divisor larger than 1. Therefore, we are looking for a multiple of 55 between 400 and 600 that is odd. One way to find such a multiple is to start with a known multiple of 55, such as 550. We can add or subtract 55 from this multiple and still obtain multiples of 55. Note that $550+55=605$, which is too large. Now $550-55=495$ which is in the correct range and is odd. Since we are told that there is only such such integer, then it must be the case that $N=495$. The sum of the digits of $N$ is $4+9+5=18$.