First, the Mayans. They and their calendar predict no such thing as the end of the world. It measures time in periods, just as ours does. Or the Jewish or Islamic calendars do. Or the Chinese calendar. The Mayan calendar just has much larger periods it can count. This cockamamy nonsense about it predicting the end of the world stems from some idiot mid-century looking to cash in on bad archeology and crackpot theories. Oh, and someone FINALLY got around to asking actual Mayans what they thought about their ancestors' calendar predicting the end of the world. The Mayans were a bit perplexed how someone could think it predicted any such thing. Again, it's a calendar. One that can also measure really long periods. That's it. That's all it does.
Second, the 26,000 year thing. That's called precession. The Earth is doing it right now. Every second, of every day, of every year. Have you ever spun a top? Besides its obvious spin, see that little wobble it does about its spin axis? That's precession and that's what the Earth does, but much much slower. 26,000 years to complete one cycle. Star charts have to be updated every 50 years because of it. While Polaris is our current north star, in 13,000 years Vega will claim that title as the precession of Earth's rotational axis slowly swings around. There will be many "north stars" along the way between Polaris and Vega and back again, but those 2 are currently the brightest along that circular path. Remember too, stars are not stationary either so eventually both Polaris and Vega will cease ever being our north stars and other stars will take their place.
Third, you hadn't mentioned it but it's another misconception that gets mixed in with Mayans and precession that grinds my gears, the Earth's magnetosphere. It goes through somewhat less predictable cycles, though we can see past cycles by measuring magnetic footprints in the rocks out by the Mid-Atlantic Ridge as the North American and European plates are pushed apart. The Earth is not going to lose its magnetosphere, the field simply weakens and becomes multipolar as a result. Instead of a magnetic north pole and a magnetic south pole, we get multiple magnetic poles. This lasts for a time until the magnetic field reorganizes itself, as it always has, and returns to a bipolar state though reversed from its previous state. Magnetic poles are NOT the same thing as rotational poles. Earth will NOT flip around, the crust will NOT shudder and break, Sol will NOT bake us alive. A weakened magnetosphere simply allows more high energy particles to reach the Earth's surface. You'll see an uptick in cancers, more electric grid failures during solar storms, and that's about it. Oh, and your magnetic compass will be utterly useless too as it won't point towards THE magnetic north pole, it'll just point at the nearest one. There will be several to choose from.
Of all the things that can kill us, Mayans and magnets and axes (oh my!) aren't among them. Rest easier. 
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