Miss diagnosed fuel pump failure

1993, Honda Accord, 4 cyl, automatic, 2 door, 266k miles. A couple of months ago I had problems with the right rear brake not releasing. I had all four brakes redone with new calipers, rotors, and pads. A couple weeks later the ABS light came on, but turned off almost immediately. I had no problems with the brakes since then.
Two weeks ago the car would turn over but not start. I had the fuel pump and cables running to the fuel pump replaced. The car still randomly would not want to start. I can hear the fuel pump kick in, so I do not think it is the problem.
Today the ABS light came on again, and then turned off. When I applied the brakes everything would be ok untill the car slowed to about 5mph then a grinding sound could be heard in the consol, and the car would shudder. After a while the problem worsened. Every time I applied the brakes to slow down past 5mph the engine cut out, or there was no response to gas until the engine cut out. The fuel pump would have to be primed by me turning the key halfway, before the car would start again. I slowed down the car on several hills without applying the brakes and it did not cut out. I could also apply the brakes lightly to slow the car to a stop without it stalling. All fluid levels are at the right level (up to date on all fluid changes).

To put it simply: Applying brakes=no power or dead engine. Slow speeds or just running the car does not kill the engine. Lightly using brakes to slow down does not cause problems either. Can the ABS failing cause engine stalling?

I know this is a mind twister. Thanks for any help that you might be able to give.