Do photons slow down this much in the Sun's gravitational field?

Well, photons always travel at the speed of light (in a vacuum and in this case between particle collisions - see below) about $3 \times 10^8 \ m/s$ and they are being slowed down in this scenario, but not the way you think and not because of the suns' gravitational field.

You should also note that the photon emitted at the centre of the sun and the one escaping at the suns surface are not the "same" photon.

Because the sun is extremely dense, a photon emitted at the core will be absorbed by another nearby proton almost immediately, and the proton will vibrate then re-emit another photon in a random direction. This happens over and over again trillions of trillions of times so that by the time it reaches the suns surface, thousands of years have passed. This process is described by what is called a random walk.

The distance that a photon can travel before it is absorbed, is given by what's called the mean free path and is given by the relation

$$l = \frac{1}{\sigma n}$$

(from Wiki) "where $n$ is the number of target particles per unit volume, and $\sigma$ is the effective cross-sectional area for collision."

As you can appreciate, the number of target particles (protons) will be significantly high making this distance extremely small, so that effectively, the photon travels a vast distance from within the Suns' core to its surface. Then it takes a measly 9 minutes to reach us!

Gravity has little to do with it. The sun is a dense plasma and particles of plasma prolifically scatter photons (both elastically and inelastically). Hence the photon does not travel from the center to the surface in a straight line.