The Corolla could be shaking for many reasons. Start by checking the wheel bearings and brakes for wear. Then check the engine to see if it running rough, causing the shake.
The engine is running rough. Could be in need of a tune up.
Check your Plugs and coils
300zx twin turbo lots of black smoke and running rough
Could be in need of a good tune up.
Running rich bad fuel injector?
If an alternator is going bad it can cause a rough idle in a 1993 Subaru Legacy. Cold temperatures in the winter can also cause a rough idle in the Subaru Legacy.
Could be in need of a tune up.
Foaming often indicates that the fluid level is too high. Maybe you overfilled the oil? Rough running could be many things. If you haven't changed plugs and plug wires lately, that could be a good place to start. FriPilot
Trail shoes are shoes for running in rough terrain. They are bulky and heavy I do not recommend running with them in places were there is not rough terrain.
There could be many reasons as to why this car is running rough. The engine may be overheated, the mount may be destroyed, it may not be getting enough fuel, or there could be a lack of oil.
It's extremely unlikely. Much more likely is rough running (usually due to engine management fault) causing a lack of oil pressure. Also, keep in mind that an excessively worn engine can cause both low or erratic oil pressure and rough running, but this will usually be accompanied by a knocking noise.