[JennyGracie (OP)]

Wendy wants to know,

Could our dramatic storms, volcanic eruptions, crop failures, and other recently extreme occurrences possibly be caused by a change in the Earth's rotation? Even a small change in speed or tilt could change a lot! Or could the hole in our ozone act like a hole in a tire, making it wobbly?

But what do you think?

[Janus]

The Earth's rotation just can't change on its own. Something would have to cause that change.  An earthquake can cause in a change of the Earth's rotation because it results in a shift in parts of the crust. The 2011 Japan quake is estimated to have shortened the period of the Earth's rotation by ~1.6 microseconds due to such shifting. The tilt of the Earth is constantly shifting due to various influences, and it is constantly slowing down by giving up angular momentum to the moon, but there is nothing different about now then any other time. 
A change in rotation or tilt  large enough and sudden enough to cause these events would also be noticeable by other means, and no such observations have been made.

[Bored chemist]

There is (or, at least was) a lab in the US whose sole activity was to measure the rate of rotation of the Earth.
So, any change would have been noticed. Indeed, tiny changes are noticed from time to time.
A really big earthquake can make a tiny difference to the rate of rotation.

[evan_au]

There is (or, at least was) a lab in the US whose sole activity was to measure the rate of rotation of the Earth.

Measuring the rate of rotation of the Earth is called "measuring the time of day".
- Most nations have a time standards lab which does this function.
- And small nations that don't have a national lab can use GPS (the GPS system is supported by an accurate time-keeping lab)
- Periodically, a "leap second" is used to bring the variable rotation of the Earth into line with the much more precise atomic time.
See: https://en.wikipedia.org/wiki/Leap_second

[alancalverd]

recently extreme occurrences

are probably not recent or extreme. The problem is that homo sapiens is leading an increasingly fragile existence, where very dense populations rely on complex infrastructures  (particularly electrical distribution grids, but increasingly overloaded sewers) and occupy increasingly marginal flood plains and semiarid regions.

It turns out for instance that the incidence of Category 4 and 5 hurricanes has not alteres significantly in the last 100 years, but the damage done by them has multiplied in direct proportion to the population of New Orleans and the Florida coastline.

[Bored chemist]

There is (or, at least was) a lab in the US whose sole activity was to measure the rate of rotation of the Earth.

Measuring the rate of rotation of the Earth is called "measuring the time of day".
- Most nations have a time standards lab which does this function.
- And small nations that don't have a national lab can use GPS (the GPS system is supported by an accurate time-keeping lab)
- Periodically, a "leap second" is used to bring the variable rotation of the Earth into line with the much more precise atomic time.
See: https://en.wikipedia.org/wiki/Leap_second

Most countries make use of the international ensemble of clocks that maintain UTC.
And it's not got much to do with the rotation of the Earth- that can get half a second out of kilter before anyone does anything.
On the other hand, there is, or was, a lab in the US whose job was to measure the rotation of the Earth. They kept records to sub-millisecond accuracy.
Without them (and others like them, nobody would know if we needed a leap second.