DropBox Cannot Connect or Sync Despite Advanced Reinstall - Need Manual Sync Solution

This morning, approximately 10 minutes after I booted my computer, I got a popup saying Dropbox was offline and to check my internet settings - the same one I get every time I start my computer.

I clicked okay and ignored it, focusing on other tasks.

About 10 minutes later, I got another popup - much larger, asking me to sign in on my browser (which popped up in the background). Apparently I was then auto-logged-in on the browser, which gave me a big blue button that said "Open Dropbox" - but the popup stayed in the foreground, asking me to go to my browser... the same one that had already logged me in.

Clicking "Go to browser" on the popup opened a new tab in my browser that indicated I was logged in and gave me the blue button to "Open Dropbox." Clicking that brought the unchanged popup back to the foreground.

At one point in this never-ending loop I attempted to "Create account" instead of "Go to browser," and entered my login information that way. I got back to the same "Open Dropbox" logged-in browser tab that went nowhere.

I closed the app and restarted it, doing other things to see if the popup would come back, but it has not, and I'm still left with the "currently offline" message with "Connecting..." showing in the taskbar over the greyed out Dropbox logo.

I don't know if this matters, but Task Manager currently shows two instances of "Dropbox Update (32-bit)" running at the same time, which is also odd.

As to your suggestion, you'd think HP's IT department would be responsive, but they can't seem to understand that I'm talking about Dropbox and not some HP-installed app. They are... unhelpful (if I'm being nice).