After some update, both post and user forms that we created previously are no longer editable. When we open them, it looks like the Post type to create isn't selected, there is no save button, and no form editor (though there is a new or previously unnoticed "Expert mode" toggle. The forms still work, and some settings, such as the success message, are still visible from within the editor screen.
You are correct, it is a plugin conflict. Disabling all of them caused the forms to work again. Unfortunately, this is a complicated site with a lot of plugins. I'll start testing them individually to determine which one is at fault, and report back here again. Thank you.
Amazingly, it was the first plugin I tried - Accordions (https://wordpress.org/plugins/accordions/). I tried rolling it back to previous versions, and version 2.1.7 doesn't cause the issue. That means that the problem occurred when it was updated to 2.1.10 (which doesn't work) and persisted in 2.1.11 (the current version).
Any suggestions on getting it to work with my forms? For now I can keep the older version, but that's probably not going to work forever.
Sorry for not closing this ticket. The accordion plugin support team fixed the issue on their end with an update, so it and Toolset's forms are now compatible again.