New Posts Not Showing For Some, Showing For Others

Here is my site: https://andrewstoeten.com

I am getting complaints from users — and sometimes finding issues myself — with blog posts not showing up once posted. Usually for me this issue is occuring in IE9 and Safari on iOS, with Chome and Firefox working well, but other readers using those browsers say that they have been having issues in them also.

I have disabled all my plugins (changing the name of my plugins directory to plugins.hold), I have manually deleted cache files in CPanel, and tried to remove any instance of old caching plugin installs (I previously used W3TC, and another plugin prior to that).

I have also switched themes (including one I installed newly) and was unable to see the changes in IE (though Chrome and FF instantly picked it up, for me, at least).

In other words, I’m stumped! Anything I can do before a full reinstall? And how best to save all my content if I do? And what if that doesn’t work!

Any suggestions will be appreciated.