Can a snake be used to remove a blockage in any drain in the house?

We have continual back-up into the bathtub of filthy water whenever we use the washing machine or drain a full sink in the kitchen. We live on one level in central Maine. We have city water but a septic tank and presumably a leach field. We had a plumber come to pump out the septic tank, but he said it was not even 1/3 full, that our problem is a "clogged leach field". I understand that may be the problem, but wonder if maybe there is something clogging the pipes. If we use a snake, would we be able to pull out something, such as a sock, that may be clogging the drains? Or to snakes just push obstacles out to the septic tank? It was suggested that we put a dye in the sink drain to see where the "gray water" comes out, to see where our leach field is and if it is working correctly. What type of dye would be environmentally safe, because we live across the road from a stream used for fishing and kayaking.