I would bet a tall cold one it is the alternator ground wire (since I have been in dry country for six months...). I know you said you checked it see if it is secure but on three seperate vehicles that failure mode for me has been the rotting of the wire inside the crimp of the ring connector that goes to the crossmember or block. The fact that it happens on acceleration and gets better on deceleration is the key.

If you doubt my diagnosis:

1) Run a heavy guage jumper from the alternator chasis directly to a known good ground.

2) That being said, the fact that it will not run on the battery alone indicates that there is no ground connection from the block to the battery. Hook you jumper cable from the ground strap mounting location on the block to a known good ground on the chasis (no the battery!) If the problem goes away, the strap or either connection from block to chasis is bad.

3) If the problem remains, hook the jumper cable from the chasis ground location for the block-to-chasis strap to the battery negative clamp. If the problem dgoes away you have a failed cable (normally from internal corrosion) from the battery to chasis.

4) Finally if that does not solve the problem, remove the negative cable entirely and connect the chasis ground to the negative post on the battery. If it suddenly works, you have a bad negative battery clamp.

After all these steps, if it still does not work you need to follow the same troubleshooting process for the positive side of the charging system. All I laid out here is the step by step fault tracking proceedure....start from the location of the issue and work backward in the smallest steps possible until you find the fault.

Onkel Udo