Most likely the problem is a bad transmission.
In order to replace it the washer has to be taken apart almost completely and you will need a helper.
Unfortunately I do not have a service manual for this washer and I'm not able to guide you through the process.
May be somebody else on this forum can help with the step by step instructions.
- The part number for the transmission is
AP2046402
- Here are the break down diagrams for the GE washer
Model WWSR3090TCWW
Gene.