Do you all really want a solution?
It's hilarious. And stupid. And VERY temporary with lots of potential issues. I am sort of hesitant to put it on here because Dropbox folks may change their stuff to counteract it.
1. Change your Mac OS Version to 10.6.8 via these instructions:
https://justindaigle.com/blog/2010/02/tutorial-change-mac-os-x-system-version/
2. DO NOT RESTART.
3. Just run Dropbox again. Log in. Let the "newer" interface resync your files. It will take longer than normal. It did not write over any of my files and even uploaded the only file I added to my Dropbox account from another computer today.
4. Delete the Dropbox shortcut in the startup list in System Preferences -> Users. It will probably just log you back out again, so not particularly a big deal.
5. With Dropbox running, reverse your Mac OS version to 10.5.8 to prevent other weird issues from happening. (By the way, with your OS version string updated, DO NOT RUN SYSTEM UPDATE.)
Problems:
If you need to resync after a restart, you will have to repeat the process. Boo. But, it does seem to sync files and even kicks out some weird plain text files relating to server addresses. However, it eventually does sync successfully.
The client app looks a little bit nice and upgraded, but it is not fully functional. It will open your Dropbox folder. It will point to the Dropbox site, but it will not show the little contextual menu for settings. It also shows files in a strange order - most recent file (in the image below) and then older files. No idea.
Crazy.
After taking that screenshot below, it did prompt me to have them automatically uploaded to Dropbox. So, parts of it continue to work just fine, but that contextual link does not work.