Wobble when I hit the breaks at high speeds 60 to 65mph

Go to an Alignment Only Shop & have them road test
while you ride along

Then have them inspect for worn parts

I would check the fuel pressure, sounds like a fuel delivery problem...possible dirty fuel filter. The vibration is probably due to a missfire. Have you had had a tune up lately? Check for vacuum leaks. I hope this helps.

If you have had any tires replaced lately check for loose lug nuts they will make the noise you have described .Please check box below if this was at all helpful.

I would call this 1 of 3 possible problems.....

This first could be a bad or unbalanced tire. Sometimes the multiply ties we use on cars separate between layers, this causes an un-natural bulge that can not always be seen, but certainly can be felt while driving. This is called ply separation. An unbalanced tire, which could be caused by a wheel weight falling off, or a shift in weight that could be cause by something like an impact break or the ply separation, would also cause a wobble that varies at different speeds. The best way to check it out without a shop visit, is to take the wheel off and just inspect it, make sure everything is smooth and even.

Secondly, a wheel bearing could cause a wobble as well. The self test for this is to jack the car up, leave the tire on and wiggle it. It should feel tight all the way around and not move up to down, or side to side. A way to test this unit while driving, which is not always accurate 100% but will give you a clue as to where to look first is drive in an area free from other traffic, when it starts to wobble, turn to both the left and right without touching the break. If the wobble goes away it is likely the cause.

Thirdly the ball joint could also cause a slight wobble. To check this, I usually jack the car up, do the same wobble test as described above, then remove the tire and inspect the joint. A good joint will have absolutely no tears in the boots. It should also feel solid at you try to move the other components that surround them.

Good Luck, we are here if you need anything further.

could be a bent rim too

How about the rear bearing? These cars do have them!

you need to jack it up and spin 1 tire at a time, use a run out guage on the tires and wheels.a high speed 60 to 65 mph is from wheel balance. low speed is a bent wheel or a bad tire. belt separation

could be a bd axle or ball joints wheel bearings are also a concideration

I would start with replacing your tranny fluid. black is no good. your tranny should hold about 7 qts of fluid. The tranny fluid runs through your radiator to help keep it cool. If your thermostat is stuck closed your eng coolant won't circulate and the coolant will get hotter inside your radiator. thus making your tranny fluid hotter and breaking it down. After time it will turn black from getting too hot. Let your car warm up to normal operating temperture then grab your upper radiator hose. It should be very warm. If not your thermostat is stuck closed.

Itcould be that the brakes installed on your car were not the Original Manufaturers (OEM) recommended pads and/or shoes on the back and front. Or only two of the brakes were replaced. when installing breaks install the OEM's or original breaks for your Corsica.
If a given set of replacement linings are more aggressive than the original, the brakes may feel grabby or upset front-to-rear brake balance increasing the risk of the front or rear brakes locking up on wet or slick roads especially if the more aggressive linings end up on the back brakes. On the other hand, if the replacement linings are less aggressive than the original linings or have less fade resistance, the vehicle may require increased pedal effort and take longer to stop than before. And if these linings end up on the front brakes, they may increase the risk of rear wheel lockup, too.