Hmm.... Now this is getting interesting! They flip it back, every time.

Apparently there's yet another element of NetworkSolutions heeldragging against transferring domain
name registration away from them.
I took a screenshot of NetworkSolution's status page for everist.org after I turned off
privacy the first time, just before 10am. (I had a hunch it might be useful.)
See attached 20151123_09-54am_NS_privacy_off.png.
It's showing 'off'.
Now at 12:00 I checked whois: shows private. Logged into Networksolutions again, and it's back to private.
I switched it to OFF (ie public) _again_.
Wow! I logged out, then logged back in to check the change stuck. It didn't. Shows 'private' again.
I set it to public **again** (3rd time now.)
Here's the text of the page with the privacy control. Ha ha... "Network Solutions feels strongly..."
See 2nd attachment: 20151123_12-13pm_NS_feels_strongly.png
 
Then it returns to the status page, showing "Registration - Private Registration Status: 	Public"
Then go to another part of their site (see 20151123_12-18pm_NS_renewal_page.png  included to show the
oddly different registration end dates of the registration and privacy option. How come they are different
anyway? And why does privacy expire before the registration? Since I think I started and always renewed both at the
same time, or if not, then privacy would have been added (for x years) AFTER the domain registration.
Weird. Or rather, cheats and liars.

Anyway, hopped around several more NS site pages, still logged in.
Return to account status page, still shows 'public'. Log out. Log back in. Now showing 'private'.
So, it's programmed to make people _think_ the change stuck, but as soon as they look away it flips back.
Thus blocking transfer of registration away from NS.

Sheesh. OK so NetworkSolutions should be called NetworkBastardry. 
What do you suggest I do?

Apart from write their shenanigans up in my blog, which I will be doing.
See http://everist.org/temp/networksolutions_sucks.txt for this text - another demo it's my site.
When I figure out how to write some nasty feedback to networksolutions I'll be doing that too.