Brakes overheating and gears locking up

you should use the highest gear possible to allow over drive to kick in, other gears will slow and limit the speed you can drive ,and stress the tranny

check brakes or handbrake are not binding. especially if the car has been sitting for a long time with the handbrake on, you release the brake lever but the sytem can still be on, after driving for a while with little braking like on a main road or freeway stop in a safe place and check your wheels with some water to see if there is steam from the brakes this means they have been binding and overheating. for handbrake drum this maybe under the car and not on the wheels. so that has to checked too to see if it hot if so the break is stuck on. i would go to a garage and have this fixed as the break pads may need replacement if they have been overheating, to ensure they work properly once break systems is fixed.

I hope this was useful.

Parking brake cables are broken, rusted or frayed. You need to replace this cables. Driving around like that will cause the rear brakes to overheat causing probably damage to rear brakes and bearings.

Well, yes the solenoids would cause the transaxle to not function right.
Are you sure the trans is not engaging, or are the brakes locked on ?
You say it shifts fine, you mean you can move the shift lever ?

The gearshift interlock prevents the shifter from being moved out of the PARK position unless the brakes are applied. The system uses power from the brake lamp switch to determine if the brakes are applied.

The first thing to do is to check your brake lights to see if they are working. If the brake lights are working, then your problem is most likely a defective shift lock solenoid.

If the brake lamps are not working, then you need to diagnose your brake lamp circuit to find out why. A common cause for the brake lamps to not work is a defective brake lamp switch. however, please be aware that there are other things that can cause your brake lamp circuits to not work. In any case, if your brake lamps are not working, getting the brake lamps to work will fix your shifter problem.

The park shift lock needs adjustment. The solenoid that locks irt inplace maybe the cause of this

although uncommon, it is possible for a failed vacuum booster to apply the brakes unintentionally. to test this, unplug the vacuum supply to the booster and plug it with something so it doesnt leak. drive the car. if the brakes dont overheat then replace the booster.

I would suggest that you adjust your rear brakes. Most likely you have drum brakes in the rear. When the rear brakes are out of adjustment (caused by not using "e-brake" when you park the car. The "e-brake" when applied keeps the rear brakes adjusted with a self adjusting mechanism.) the front brakes have to do ALL of the work to stop the car. In this case going down a steep grade for a long time simply overheated the front brakes. When the front brake pads overheat they get "glazed" and don't want to work anymore. Most likely reason for brake pedal going to the floor is the brake fluid was boiling. Hope this helps. Good luck.

Check brakes first..frozen.calipers or internally failed brake flex hoses may be causing brakes to remain on. Additional load may be causing engine to overheat. Likely if this has gone on for any length of time rotors have overheated and need to be replaced as well.