The devs at Bethesda were nice enough to contact Arthmoor about the dialogue bug we had all been experiencing. After some private testing on his part, and later mine, we determined that it was indeed the source of most of our issues.
The problem is that we need to generate something called a SEQ file. This file tells the game engine which quests you have start-game enabled, and possibly which ones you have altered. Generating this file should result in all of your dialogue and scenes working properly once again, just as they did pre 1.6.
How do you generate this file? There are several ways. You can do it via the windows command prompt, via Steam launch options, and once released, via TESVEdit, which will have a built in button.
I've made a video with instructions on how to do it via the Steam launch options: http://youtu.be/Ai3txtvoSbM
Here are the written instructions--
If you wish to use the Steam Launch Options:
1. Open the main Steam window, and go to the Library tab.
2. Click the dropdown, and select 'Tools'.
3. Right click on Creation Kit in the list.
4. Select 'Properties'.
5. Click 'Set Launch Options...'
6. Type in -GenerateSEQ:PluginName.esp Where PluginName is the name of your plugin.
7. Click 'Okay'.
8. Open the Creation Kit. It should show the splash screen, and load some things. This may take a few minutes. When it's done, the Creation Kit will simply close.
9. Go back to 'Set Launch Options...' and remove the line we just typed in, allowing you to resume using the CK normally.
10. Navigate to your Skyrim folder, just above the Data folder.
11. You'll have a file called PluginName.SEQ. Move it to Data/SEQ. If that folder does not already exist (It shouldn't) create it.
12. You should now have "Skyrim/Data/SEQ/PluginName.SEQ" and you're good to go.
If you'd rather use the Windows Command Prompt:
1. Open CMD.
2. Navigate your way to where Skyrim is installed. (By default: C:\Program Files\Steam\steamapps\common\skyrim)
3. Type in the following command: creationkit.exe -GenerateSEQ:pluginname.esp (Again, where pluginname.esp is the name of your mod file)
4. The CK will open just like step #8 above.
5. Repeat steps #10-12 above.
Now, there have been a few scattered reports of this not working for some people. It seems a bit iffy. And I mean generation, once you get the actual file, we've had nothing but positive reports of it working and fixing most/all dialogue related issues.
Some notes from Arthmoor:
The SEQ file will need to be regenerated any time you add a new start-enabled quest to your mod, delete one from the mod, or toggle the "start game enabled" flag on a quest in your mod. So it's important to remember this.
When I tested this, the command line option only worked once and has failed to ever work again. The Steam method works without fail each time, but it's a pain.
Good news is, once the developers cleared Alexander and I to share this knowledge, I contacted Zilav (TES5Edit) and he'll be including support to generate this file through TES5Edit as well.
The file itself is nothing more than a series of form IDs for each quest in your mod that's been flagged as start-enabled. It's encoded though, so you can't just type it out by hand.
TES5Edit will be the easiest method for generating this file since it will place it in the proper folder for you, rather than the CK simply dropping it in the same place as creationkit.exe.
Remember to distribute this file with your mod along with everything else! If you do not, people will experience the 'dialogue bug', and your mod will likely not work for them at all.
Also remember to regenerate this file whenever you alter additional quests, or add more of your own.
Thank you to Joel for reaching out to us with this solution! And, to the devs for their abrupt increase in activity.

AJV