In addition to @Stephen_C’s very useful suggestion, remember to remove the license from your copy of DT (DEVONthink 3/License…) so as to be able to re-register your new installation (in some cases that may or may not be necessary, depending on how DT have linked the license to the device. But it’s easy to do, so I recommend it, unless you only recently registered that seat, in which case I suggest you first check with DEVONtech). The same recommendation also applies to other software (e.g. Adobe CreativeSuite or similar).
Also note that Jim’s instructions in the above named thread would not copy any DT3 scripts - they are to be found in /Users/yourusername/Library/Application Scripts/com.devon-technologies.think3 - that is, of course, only relevant if you have written any scripts yourself; the standard scripts will be reinstalled automatically.
Furthermore, if you use the search method described by Jim to look for your databases, you will not find encrypted or open databases. You should close any databases before copying them (the same is true of DT itself, which should be closed before copying any settings, scripts etc.) and include .dtSparse in your search. I recommend you check to make sure that all databases really have been found - in theory it is possible that if one isn’t closed properly it would have a .sparseimage ending. Of course, ideally, you’ll know where you keep your databases and have no problem copying them
Then, for good measure, I recommend deleting the (old) Abbyy folder here /Users/yourusername/Library/Application Support/DEVONthink 3/Abbyy (which you will have copied back to your new installation as per the instructions in the thread above) before reinstalling DT; I had some trouble with system protection denying access to the files in that folder; the OCR engine (assuming you have Pro or Server) will be downloaded again if you select the appropriate Add-On in your new installation.
Don’t forget to reactivate the Mail plugin, if you use it.
And one last thing: I seem to remember that despite having done all of the above, and following Jim’s instructions, a small number of preferences were not copied (or overwritten on reinstall). As I had a second device with an identical setup, it was easy just to go through the preferences and put things back the way I wanted them. If you don’t have a second device, consider screen shots of the preferences, especially if you have taken time to set things up just the way you want them. Of course, maybe things have changed with 3.6.1 and the preferences will all be preserved. Feel free to post back here and let future generations know how things went