I had this same problem with a Moto X. Yesterday I finally got it to read aloud simply by toggling bluetooth on, even though I had nothing to pair it with. Then it worked great. Until I quit driving, that is, after which it still kept announcing EVERYTHING. Power cycled - still announced. Uninstalled Dragon - still announced. Got on phone with tech support and cleared the Assist app data - still announced everything. Cleared all Google app data - still announced everything. Later on while ending a drive I specifically clicked "I'm not driving" and it went back to announcing nothing anymore (yes, even when driving). So my experience has been an all or nothing - Assist detects I'm driving pretty well, but the hands-free announcements seem to be totally independent of that.
Last night I worked with Level 2 support and he ended up clearing up some more app data and today it seems to be working OK. But now it seems to have trouble determining when I stop driving. The L2 guy said he thought it was having trouble with the GPS part of it. Said he'd look into it some more and call me back this evening. If I get him on the phone again I'll ask him specifically what he cleared to at least get the announcements to work properly today.
So today the Moto X properly knows when I start driving and announces properly, but when I'm done I need to tell it I'm not driving anymore and it will silence the announcements as it should. Would like to have to not click that, but compared to the "all or nothing" I was seeing before I'll take it.
But what bugs me most is that this feature is one of the main selling points and one of the main reasons I chose this over the LG G2. If they can't get this major feature right, I'm wondering if I should switch while I still can...