The lid switch actually is the combination of two switches: the lid or PC switch (#5 on the
diagram) and the check switch (#17).
The water valve gets power through the PC switch while the motor can be powered up through both switches only (
wiring diagram).
Looks like the check switch in your washer does not work.
You have to check the lid switch assembly to figure out what has to be done to fix the problem. Some time it can be just the new check switch, some time the whole assembly has to be replaced.
- The check switch
Part number: AP4023948 
- The lid switch assembly
Part number: AP4008524 
Here are the breakdown diagrams and
Replacement parts for MAYTAG LSG7806AAE Stacked Laundry - Gas | AppliancePartsPros.com
Gene.