Would we have more than 8 minutes of light, if the sun "went out"? The common theory is, that if the Sun "shut down", we would see the light for eight more minutes (the time that it takes the photons to reach the Earth). 

However recently I have read that photons need around 100 000 years to reach the Earth, since the reactions are happening at Sun's core, and gamma rays can't leave the Sun without interacting with other particles, unlike neutrinos for example.

Is that theory correct? If the Sun's core "shut down", would we still receive photons (light) for another 100 000 years, with only neutrinos disappearing immediately?
If nuclear fusion were to suddenly stop in the centre of the Sun, then the only clear signature we would have of this is the lack of detectable neutrinos received at Earth, starting about 8 minutes after the reactions ceased. The Sun however would continue to shine for tens of millions of years at roughly its current luminosity.

The power source is not "stored" photons. The Sun itself would simply resume the slow gravitational contraction that was halted about 4.5 billion years ago when nuclear reaction rates at the centre were able to increase sufficiently to supply the radiative losses from the surface of the Sun.

The characteristic timescale for the contraction is about
$$\tau_{\rm KH} = \frac{GM^2}{RL},$$
which is 30 million years. i.e. The Sun has enough gravitational potential energy to supply its current luminosity for tens of millions of years.

While this is happening, the Sun would approximately maintain its current luminosity, but decrease in radius, meaning that its surface temperature would increase.

Once the Sun had contracted to a few times the size of Jupiter (so about 30% of its current radius), the contraction would begin to slow, because the electrons in the core become degenerate and the pressure increases with density by more than expected for a perfect gas. The slowing contraction decreases the rate of potential energy release and hence the solar luminosity. The contraction continues at a slow rate until the Sun becomes a hot "hydrogen white dwarf" a few times the size of the Earth, which then cools to a glowing cinder, with no further contraction, over billions of years (see What would the Sun be like if nuclear reactions could not proceed via quantum tunneling? for some more details).

Even if you were to not allow the Sun to contract, it would take some time to radiate it's thermal energy. This timescale is approximately
$$\tau_{\rm therm} \simeq \frac{3k_B T M}{m_H L},$$
which assumes the Sun is a perfect gas of protons plus electrons, with an average temperature $T$. If we take $T =10^7$ K and the current solar luminosity, then $\tau_{\rm therm}=$ 40 million years.

On the other hand, if your scenario is just that light from the Sun stops being emitted, then of course it goes dark on Earth about 8 minutes later.