Moved site, now logout problem when leaving a secure page

This is a problem that relates to the back end of the site. I build the site on one domain. Moved it to another domain.

Now there is a problem getting logged out when on the front end of the site. It happens when going from a secure page, like the checkout, back to a non secure page.

Another symptom is that when trying to delete a plugin, WP asks me for my FTP credentials.

Another weird thing is that Chrome keeps putting me into https for the WP admin.

Note that this is a WooCommerce site. It’s set up to force Secure on the checkout page, and force un-secure when leaving a secure area. That should not cause this problem though, and I’ve verified that it doesn’t cause this problem on my other Woo site.