It seems to be because two variables for the gravity equation that I used (G = 1, M1 = 1000) were not best for the scale of my simulation.

I'm no orbital physicist, so I can't explain WHY this fixed it exactly, but I changed the Gravitational Constant G to 5 * 10^-3 and the mass of the Earth to 1 * 10^9. I guess this setup more closely resembled real life scales, so gravity needed less time to act correctly? This change led to objects' perigees and apogees staying closer to their original position for longer; less ridiculous orbital precession.

There's this value called the Standard Gravitational Parameter (SGP). It's defined as G(M1 + M2). Plugging in real life values, this number comes out to 3.986 * 10^14. Using my original values, it came out to 1000. I then thought that manipulating my G and M1 so that it is closer to the real SGP would result in basically zero orbital precession but deviating from my fixed SGP of 5 * 10^6 in either direction led to more orbital precession.

So, that answers my question, but I do not understand the mathematical reasoning.