Can the fuel on the wing cause the aircraft to destabilize?
Yes it can. Source: I'm a pilot.

Most aircraft (even many small general aviation ones) have balance parameters where the fuel quantity in each side cannot/should not exceed that on the other side by greater than a certain amount.

Note that this is usually caused by fuel consumption in flight, not sloshing as you asked.

Also, (as in your question) fuel can slosh between tanks in most aircraft through the vent line that often connects one tank to another such as in light Cessnas. However this is usually only a small quantity and furthermore the aircraft is usually in level flight so it's not really a problem.

Fuel sloshing inside of one tank could potentially imbalance the plane a bit but I've never ran into any real effects of this nor heard anyone mention it. Furthermore, most aircraft have some sort I baffles in the tanks (usually wig ribs) that prevent much sloshing, whether by design or not.