I'll take a guess at this mystery.

If the new seats are slightly shorter than the original seats, the distance that the stem needs to travel is increased. This would normally work fine............if the stem has that travel available. If the stem is limited on travel, it runs out of threads just before the washer contacts the face of the seat (maybe .020" short) and it allows the flow that you describe.

Put some bluing on the face of the seat and close the stem. See if the bluing transfers to the entire washer face. Then you'll know.

The only other possibility is the face where the seat contacts is totally corroded and the threads are also shot. This should allow a drip............I cannot imagine that it would allow a stream.