You're in luck
The Alvarez Hypothesis, which is the source of our common belief that an asteroid killed the dinosaurs, has a report from some of its supporters that claim the Earth's axis did, in fact, shift due to the impact.
in a 1953 publication, geologists Allan O. Kelly and Frank Dachille analyzed global geological evidence suggesting that one or more giant asteroids impacted the Earth, causing an angular shift in its axis, global floods, firestorms, atmospheric occlusion, and the extinction of the dinosaurs. (Source)
Further, there are articles that describe how the Earth's axis can shift due to impacts (for example...).
The question is, are you looking for "realistic" or "suspension of disbelief"?
I'm not a fan of "realistic." I think too many writers get bogged down in the details of "realism." It beats me why. There may only be two people who ever read their work who might know they were not completely "realistic" — and if the story's good, they won't care.
So, suspension-of-disbelief says you can use an asteroid impact to shift the axis. Note that a direct impact will shift the axis very little. An oblique impact will shift it more (and possibly knock a new moon into orbit). Where the asteroid impacts, and at what angle, have a lot to do with how much the Earth could believably be rolled over.
Further, you could use a series of impacts, say days or weeks apart, that achieve the axial shift and kill the dinosaurs without the trouble of turning the Earth's surface into soup.
But you have other alternatives
Another alternative is one or more earthquakes. While our current evidence suggests they won't move the axis a lot, what do you care? Think of earthquakes in terms of slamming down your garage door. Slam that tectonic plate against the others hard enough, and you shift the axis, screw up the weather, and kill a whole lot of dinosaurs.
Yet another is volcanism. Not any ordinary volcano will do. You'd need something like the entire Yellowstone Caldera going up like Mt St. Helens. This solution might be the most believable for keeping the tectonic plates intact.
If you want something more exotic, let's have a micro black-hole wander through the system
Of course, an external gravity source could believably shift the Earth's axis. Let's have a small black hole wander through the solar system. It passes Earth at just the right place to jerk the planet around, shifting the axis. The consequences would definitely kill dinosaurs!