Let the two consecutive odd numbers be $n$ and $n+2$. Consecutive odd numbers are always coprime, meaning their greatest common divisor (HCF) is $1$. For coprime numbers, their Least Common Multiple (LCM) is equal to their product. Given that the LCM is $8463$, we have $n(n+2) = 8463$. This is a quadratic equation $n^2 + 2n - 8463 = 0$. We can estimate $n$ by taking the square root of $8463$. $\sqrt{8463} \approx 91.99$. Since the numbers are consecutive odd integers, they must be close to $91.99$. The two odd integers closest to $91.99$ with a difference of $2$ are $91$ and $93$. Let's verify: $91 \times 93 = (92-1)(92+1) = 92^2 - 1^2 = 8464 - 1 = 8463$. Thus, the two numbers are $91$ and $93$. The bigger odd number is $93$.