Thanks Michael and Simon!
The login issue seems to be a conflict with the MemberPress plugin. So I guess I need to dig into a possible solution with them.
The logout issue was caused by a redirection plugin. There was a home page redirect that was causing the problem with the Discourse logout.
Michael – really appreciate your help and for the advice on plugins might be suspects. It saved me a ton of time because those were the first two plugins I checked.
Simon – thanks for telling me about the WordPress Health Check plugin. I hadn’t heard of that before but I’m going to install it now and use it for troubleshooting in future.