My best guess is that, as I cross the surface of the sun, I start accumulating solar mass on the opposite side of me from the center of gravity that’s attracting me, and that prevents the formation of an event horizon. Is this reasoning correct?
Yes, that reasoning is correct.
The simplest solution to the Einstein field equations in General Relativity is the Schwarzschild solution which
describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentumof the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun.
As balkael mentioned, the Sun's Schwarzschild radius, $r_S$, is approximately 3 km. That means that if the Sun's mass could be compressed down to a sphere of $6\pi$ km circumference it would be a black hole. But that doesn't mean that anything special occurs at 3 km from the centre of the uncompressed Sun.
Side note
Newtonian gravity is a very good approximation at distances from the centre of mass that are large compared to $r_S$. At Earth's orbital radius, the difference between Newtonian gravity & GR is minute. Even at Mercury's orbit the difference is rather small. One of the early triumphs of GR is that it correctly predicts the anomalous apsidal precession of Mercury's orbit. According to Newton, the major axis of Mercury's elliptical orbit (aka the line of apsides) would point in a constant direction, if the solar system consisted only of the Sun and Mercury, but due to the gravity of the other planets (and because the Sun isn't a perfect sphere) the line of apsides slowly rotates, as shown:
From Wikipedia:
Mercury deviates from the precession predicted from these Newtonian effects. This anomalous rate of precession of the perihelion of Mercury's orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier.
The total precession is only 574.10 ± 0.65 arc-seconds per century. The anomalous precession due to relativistic effects is only 43 arc-seconds per century. That is 43 / 3600 degrees.
I mentioned earlier that nothing special happens at $r_S$ in the Sun. That's because when you go inside a spherically symmetric body the mass above your head exerts zero gravitational force on you. In Newtonian gravity, this is due to the Shell theorem, as G. Smith said. It's also true in General Relativity, due to Birkhoff's theorem. So all of the Sun's matter that's more distant than $r_S$ from the centre cannot create an event horizon.
If you could somehow compress that matter sufficiently then a black hole would form, but no known process can do that. As far as we know, the smallest black holes that can be created in a type II supernova explosion have a mass around 3-5 $M_\odot$ (solar masses), with the progenitor star having a mass around 20 $M_\odot$.
So density is only of indirect importance, the main thing is to get enough mass within the Schwarzschild radius. Actually, it doesn't have to just be mass, all forms of energy contribute to the stress-energy-momentum tensor which is the source of spacetime curvature.
