She doesn't recognize that her thought processes have become flawed; she forgets what she's told us, she becomes argumentative when we don't agree with her. It's causing stress and we'd like her to leave the organization but don't want to push her out. Any suggestions would be welcome.
If dementia is setting in, even getting to your location may be more than she should do. We had a similar situation in one of my volunteer positions. The head of the group asked me for advice. When I heard what was going on I questioned the ability of the person to drive to the location. This got the group head nervous. Someone spoke with the volunteer about the work becoming too strenuous and asked if he would like to take some time off. He agreed and ... needless to say, hasn't returned. It is sad if it is dementia. Alternatively, perhaps just a change in the type of assignment may work.
Maybe something in her life has changed and is stressful, this can cause behaviour like you describe. Once you find out what is going on, it will help you decide how to move forward.
Offering her some time away from her volunteer responsibilities, so she can focus on whatever else is troubling her, is the least pushy way I can think of going about this.