Understanding Feldman's VSS with a simple example
It has to do with which modulus you use.  You did all your arithmetic modulo 11.  However, when using Feldman's VSS, you gotta use two different moduli (using each one in the appropriate spot).  In your example, you shouldn't do all arithmetic modulo 11.  Instead, you should be doing some arithmetic modulo 11, and some arithmetic modulo 5 (the order of $g$ in this case).  If you do that, everything will work out.

In general, you need to pick primes $p,q$ such that $q | p-1$.  Then, some arithmetic is done modulo $p$, and some arithmetic is done modulo $q$.  In particular:


The polynomial $f(x)$ is treated modulo $q$, so when you compute the shares, you need do arithmetic modulo $q$.
The commitments and all computations with the commitments are treated modulo $p$, so when you verify that you got the correct share, you work modulo $p$.


The reason we do it this way is that when doing a computation like $3^{15}$ (modulo $11$), we can reduce the base modulo 11, but we have to reduce the exponent modulo 10.  Roughly speaking, commitments are in the base, whereas the value of the polynomial (the shares; the coefficient of the polynomial) are in the exponent -- so you gotta use different moduli for these two different kinds of values.

We can modify your example to take this into account. We could take $p=11$, $q=5$, and generator $g=3$ of the subgroup of order $q$ of $(\mathbb{Z}/p\mathbb{Z})^*$.  However, you can no longer have the polynomial $f(x)=5+3x+8x^2$: the polynomial is interpreted modulo $q$, so all coefficients have to be in the range $0..4$.  This means we'll need to change things a little bit.

So, here's a corrected example.  You could use the polynomial $f(x)=0+3x+3x^2$.  Since $f(1)=1$, you'll get the share $s_1=1$.  The commitments are $3^0=1$, $3^3=5$, and $3^3=5$.

For verification of the correctness of the share, we first compute $3^{s_1}=3^1=3$.  Next, we compute the check value $1 \times 5^1 \times 5^{1^2} = 3$.  You can see that $3^{s_1}$ is equal to the check value, so everything verifies, and the share is correct.