The orbit hasn't changed (except through a different Milankovich cycle that I believe takes 112,000 years or something like that). What you may be thinking of its that Earth's orbital eccentricity changes on this ~100,000 year timescale.

When it comes to precession, what is happening is that the Earth's rotation axis is spinning around in space pointing in different directions on a 26,000 year cycle. This drift of Earth's spin axis is not related to its orbit.

Right now, the Northern Hemisphere (NH) spin axis is oriented toward the Sun in June. This means on the June Solstice the spin axis is most directed toward the Sun from the point of view of the NH. This is why the NH experiences summer in June.

Over the next 13,000 years, Earth's rotational axis will revolve (precess) to the other direction in space, and be oriented about 23.5 degrees in the other direction in June. This means that when Earth orbits to the December position, then the axis will orient the NH toward the Sun. So the date on the calendar of the seasons will be effectively flipped. That's it. There's nothing about the orbit that changes from precession. The seasons will drift through the calendric dates of the calendar though.

Finally, since the Earth's orbit is eccentric, and the Earth is physically closer to the Sun by 3% in January, this means that if December in 13000 years is Summer for the NH, then the NH will also get 3% more sunlight in summer.