The house was built in 1968, I had it snaked and was told there was no obstruction, so I piugged the drain and now its comes up the shower drain

Whoever snaked your drainage missed the blockage. If it came up out of the floor drain then the blockage's is downstream from it. Could possibly be reached through the outside clean out. Call him back and have him do the job right. Good luck, Tom

Hey Speedball, I have had my drain snaked many times and it never fixes this problem. It has been going on since we bought the house which was 19 years ago. I wonder is it possible that the drain pipe is to small for the discharge and cannot handle the flow, because as soon as the washing machine quits draining its gets sucked down real quick?

The reason the washer backs up out of the floor drain is because it discharger with great force and volume. You have a partial blockage down stream from the floor drain. I suspect it's in the line to the street, perhaps roots> To check your sewer line locate the house clean out and remove the cap. Now cycle the washer as if you had a full load and watch the opening. If water backs up then your problem's in the line to the street. Let me know what you find. Good luck, Tom