Transmission - trouble moving off from standing start

I have a 1994 ED Falcon, Automatic, 6 cyl 4.0L.
When I start up the car and put it in D, it does not engage gear - car revs and makes a nice whirring sound and goes almost nowhere before thumping into gear and away I go. Once going, it behaves - shifts ok and smooth etc. However, if I for example stop at an intersection, the car doesn't want to get going again - same problem of not engage a gear I think. If I drive it like a manual, it does select the gears and behave, although not from standing/neutral start. I can mostly keep it going by downshifting through 3 2 1 as I slow down to stop, then leave it in 1 while waiting to take off.
The transmission bands have been adjusted and seem to be in manufacturer spec, but doing this made no difference to the cars behavior. Can you tell me what may be causing the problem, or where I should be looking to find the problem and fix it? Thanks.