The OP asked me to comment.
The most important step is to remove the DEVONthink databases themselves from the Dropbox folder. I would treat this quite simply, by the following steps:
On each Mac:
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Quit the DEVONthink application. (Avoid copying open databases.)
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In the Finder, Command-click on the DEVONthink databases (their filename extension is .dtBase2) to select all of them. (Command-click allows non-contiguous selection of multiple files.)
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Hold down the Option key while dragging the selected files to copy them to a different location, the Documents folder. (DO NOT YET DELETE the database files from Dropbox!)
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Launch the DEVONthink application. If the databases still located in Dropbox were automatically opened, close them.
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In DEVONthink, choose File > Open Recent > Clear Menu. (This will clear DEVONthink’s memory of the previous location of the databases.) Now open each of the databases copied from the Dropbox location to the Documents location, so that DEVONthink knows their new location.
Repeat those 5 steps on each Mac that holds copies of the database in Dropbox and that are to be used for database work.
Afterwards, select all of the database files (their filename extension is .dtBase2) that are stored within the Dropbox folder, Control-click and select the option to move them to the Trash.
Run Tools > Verify & Repair on your databases on each Mac to check for any errors, and run Tools > Rebuild Database if errors are reported. If Orphan files are shown in an Orphans group, refile them into your database organization.
Now your databases are safely stored for use and you will be able to Sync them among your Macs.
Although multiple steps are involved, they are standard OS X procedures for copying files and can be quickly performed on each Mac.