1987 Buick Lesabre stalls and then is hard to start.

I have had a similar problem with my 1994 Buick LeSabre. So far I have diagnosed that it might possibly be the Emissions Control System. Check your charcoal canister. It is located under the hood in the very front, driver side corner. It is a large black cylinder that sits down into a hole. Make sure the fuel line is connected into the fuel port and the smaller vaccuum line is connected into the other port. On my 94 I have three ports and I am unsure of what port to use for the vaccumm line because one says purge and the other says air. I believe its just the purge port and the air port is left open. I found my lines unconnected. But other systems for the emissions might be causing the problem. For example I saw another post on here with someone and similar symptoms to you and myself and they solved the problem with a new camshaft sensor or crankshaft sensor. Try a new Mass Air Flow sensor as well.

The hard to start and possibly stalling problem could be the cause of a bad "Crankshaft Position Sensor"

They are around 20 dollars if I remember right; I had one of these replaced on my 87 lesabre which had similar problems.

The sensor is located by the harmonic balancer so you would need to take the right front wheel off to get to it.
There will be two sets of wires (around there) one is the camshaft(upper one), other is the crankshaft (should be the easier one to get to)
Also you may need to remove the engine belt to get to the bolts.