Reddit reviews Explanatory Supplement to the Astronomical Almanac

Earth rotates (nearly) freely in space around a principal axis of rotation which, as you say, we know as the north-south axis. If you were to apply some torque to the Earth to change that, the rotation would change to some other axis. In other words, you wouldn't really see any more complicated behaviour than today, just a different axis, and there would be new "north" and "south" poles.

That's the boring answer. Now for something slightly more interesting.

Now as it happens the Earth does actually not rotate freely. It is subject primarily to tidal forces from the Moon and the Sun, but also a few others. This causes the axis of rotation to change over a period of roughly 26,000 years, an effect known as axial precession or precession of the equinoxes. The apparent north pole, currently in the vicinity of the star Polaris, will trace a circle over 26,000 years. The center point of that circle is the "orbital" north pole, i.e. the axis around which the Earth orbits (not rotates). This looks similar to how a spinning top sometimes slowly spins around its own axis of rotation, if you set it spinning with a slight tilt on a surface.

There are other changes to Earth's rotation too, they are called nutation, the principal effect of which varies over about an 18 year period.

So yes, it is possible to have such complicated rotation and in fact it already happens. However I do not wish to speculate on how exactly precession and nutation would look like if you modified the Earth's primary axis of rotation and/or angular momentum.

Source: Explanatory Supplement to the Astronomical Almanac.

If the characters wake up with an astronautical book or data base, then the positions of the moon and planets would be the best bet for telling the date and location. The stars would give them only the latitude, not the longitude. Although the astro almanac is recalculated for each year, it should be accurate enough to project the equations a few hundred years into the future. If you need a few thousand years, then you need a long-term epidermis, like JPL DE431 (covers years 13201 BC to AD 17191). Of course, you need a computer to run it on.

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https://en.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris