Or the other way round, yes, you are looking for continuity between the switch wire and ground with your meter on the Ohms scale so it doesn't matter which way round you connect the leads. Be aware that the diagram in the ETM, the one above, is opposite to normal convention which would be 1 for closed and 0 for open, and you are looking for 1 being open and 0 being closed. So with it in Park, X switch would show closed, a complete circuit, while Y and Z would both show infinity.
If it has been filled with water, that would explain it, you wouldn't see infinity on the open switches but a high resistance. When in limp mode, the display on the dash will go off and not show what gear you are in. The breather was once a U shaped piece of tube so the open end of it pointed downwards to stop water getting in. Mine was missing so it was replaced with a bit of washer hose.
The part number you have quoted is for 98 onwards, it is STC2944 for earlier cars which should have a black plug, the later one has a grey plug.