Will the equinoxes and solstices switch places in 13,000 years because of the precession of the Earth's rotation axis?

No, there will be no swapping of the seasons and the months of the year.

The precession of the equinoxes of the Earth is a motion that causes the axis of the Earth's rotation to remain FIXED at an angle of 23.5 degrees, however, it rotates along a great circle with a period of 26,000 years. The result is that the two points where the equator of the Earth intersect the ecliptic plane, the vernal and autumnal equinoxes, precess westward along the ecliptic by 360 degrees per 25,800 years or 50.26 seconds of arc per year. This is also equal to 0.125 seconds of arc per day or 0.008 seconds of time, so that each day the synchronization between sidereal and solar time slips a bit. Currently, the vernal equinox which heralds the beginning of spring occurs in the constellation of Pisces, but it is slowly moving towards the constellation of Aquarius and will arrive there in a few hundred years or so.

The seasons of the year are produced by the tilt of the axis of the Earth, and this tilt is not disturbed by the precession, but remains exactly the same with respect to the ecliptic plane. Currently in the northern hemisphere, the Earth is tilted TOWARDS the Sun by 23.5 degrees when the Earth is at its farthest from the Sun ( aphelion ) in June, and we experience summer. In the winter it is tilted AWAY from the Sun today. Because our calendar year and its seasons are tied to when the equinoxes occur, it automatically keeps up with the precession, so that in 13,000 years we will have the following situation: The axis of the Earth will be tilted TOWARDS the Sun by 23.5 degrees when the Earth is closest to the Sun ( perihelion) in the northern hemisphere in JUNE, and tilted AWAY from the Sun when it is closest to the Sun in December. Each day, our calendar is gradually 'precessing' in time by 0.008 seconds to keep up with the new locations of the equinoxes and solstices so no matter where we are in the precession cycle ,winter will always happen in December, and Summer in June. BUT because in the northern hemisphere in 13,000 years we will be closer to the Sun for our summer, and farther for our winter, the severity of these seasons will be slightly greater.

What will also change is the constellation that the summer solstice will be in. In 13,000 years the summer solstice will travel 1/2 of a full cycle around the zodiac. Will we still celebrate winter in December and summer in June in the northern hemisphere? Yes, and we will also see Orion as a summer constellation in 13,000 years. Precession affects the background constellations against which the Earth-Sun motion plays itself out. It does not affect the months during which the seasons occur, because these are constantly being updated to keep the vernal equinox in March etc.