To ensure that this isn't being caused by a 3rd party plugin I would recommend temporarily disabling all the non-toolset plugins and trying again to see if the issue still remains.
Okay, I deactivated all those I felt comfortable deactivating (please let me know if I should do more safely!) But no difference - still slow as molasses. Last test was 16 seconds to refresh.
Next?
Following is list of deactivated - and still active plugins.
*Deactivated plugins:*
Email Subscribers & Newsletters
Search shortcode
W3 Total Cache
WP Ultimate CSV Importer Pro
1&1 WP Wizard
Relevanssi Premium
Show-Hide/Collapse-Expand
Nav Menu Roles
Contact Form 7
Hide Admin Bar
Google AdSense plugin