Car will not stay running for long before it shuts off, again

Hold on. A fault code did not say the whole transmission is bad. Fault codes never ever say a part is bad or to replace one. They only indicate the circuit or system that needs further diagnosis, or the unacceptable operating condition. If you reply with the exact code number, I can look up the definition for you, and possibly offer some suggestions.

The intermittent stalling you described is most commonly caused by a failing crankshaft position sensor or camshaft position sensor. If you are lucky, a diagnostic fault code will set to indicate which signal is going missing, but sometimes the engine stops rotating too quickly for the code to set. These two sensors also commonly fail on any brand of car by becoming heat-sensitive, then they work again after cooling down for about an hour. The biggest problems occur when stopping a hot engine for a short period of time, such as when stopping for gas. Heat from the engine migrates to the sensor and causes it to fail. Typically when driving, natural air flow keeps the sensors cool. I looked up the sensors for a LeSabre. The crankshaft position sensor for that model is down lower where heat is less of an issue than for the camshaft position sensor.

When there is no fault code related to either sensor, you can use a scanner to view live data. Most show those two sensors with some indication as to whether their signals are showing up during cranking. When one signal is missing, that sensor is the cause about half of the time. First we have to rule out wiring and connector terminal problems. In this case when the vehicle sat for so long, wires chewed on by mice is a real good suspect.

P0440 - Evaporative Emission Control System Malfunction.
P1870 - Transmission Component Slipping.

The list of suspects for code 1870 depends on the car model. Which model do you have?

Slippage is detected while the car is at cruising speed in the higher gears. The cause can be due an internal clutch pack problem inside the transmission, a hydraulic system problem, or an electrical problem that controls the hydraulics. You are not going to be happy if you replace the transmission and have the same problem.

The first thing to look for that is a common cause of this code is low or contaminated transmission fluid. If air gets sucked up into the pump, it can be compressed and will not allow the clutch packs to apply fully. That will result in slippage, and code 1870. The solution to this is to fix the leak and fill the fluid.

A real common cause of this code on some models is a cracked clutch disc inside the torque converter. It is common enough that a lot of people automatically replace it for this code and fail to eliminate the other possible causes first. If the slippage is caused by the lockup clutch in the torque converter, replacing the entire transmission is a waste of money and is not necessary.

You're describing two different things. A lot of people confuse this, so to be sure we're clear, when you say the engine "doesn't crank again... ", That means the starter does not spin the engine. That could be due to a run-down battery, bad cable connection, and things like that. A failed generator could let the battery run down to the point the engine stops running in 500 feet, but the battery could recover enough in a few hours to crank the engine fast enough to start again. That's a common charging system problem that GM owners are REAL familiar with.

Where some people confuse the description of the symptoms is when they think "crank" means the engine is running. The term "crank" comes from the days we used a hand crank to spin the engine to get it to run. Today, the starter cranks the engine, then if everything is working, the engine will start and run.

If you really have a starter that won't crank the engine at times, I would start with a charging system test. You can start that yourself if you have an inexpensive digital voltmeter. Measure the battery voltage with the engine off first. You will find 12.6 volts if it's good and fully-charged. If the battery is run dead, it will measure close to 12.2 volts. If you find 11 volts or less, it has a shorted cell and must be replaced.

Next, measure the battery's voltage again with the engine running. Now you must find between 13.75 and 14.75 volts. If that is low, suspect the generator. If it is good, that only means it is okay to do the rest of the tests, but you need a professional load tester for that. Specifically, we want to know the values for "full-load output current", "charging voltage", and "ripple voltage". A few testers provide ripple voltage as a value, but most just list it as "low" or "high".

Now, to shift gears, if the starter is always cranking the engine okay, but the engine doesn't run, go back and reread the last two paragraphs in my first reply. These sensors also commonly cause engine stalling on any brand of car. Most commonly they fail while a hot engine is stopped for a very short period of time, as in when stopping for gas, but they can also fail in the way you just described. It's not uncommon for no diagnostic fault code to set to direct you to the circuit that needs to be diagnosed. That makes it more difficult to identify the cause, and it is almost impossible without a scanner.

None of this relates to the 1870 fault code, including the stalling. There might be some other problem we have to pursue later, but nothing in the transmission will cause the engine to stall while you're driving. Of the list of possible causes for that code, the most common one would simply cause a small decrease in fuel mileage.