You can output your movie for authoring in another program by selecting Export & Share/Devices/Computer 720x480 and outputting to the MPEG-PS. This will output your movie only of course and will not include your DVD markers or menus. Those are elements you will add in your DVD authoring program.
Beyond that, I've always gotten virtually as good as the original quality from my DVDs (except, of course, for being standard resolution instead of high-def), so I can't speak for your quality issues. But this should be true for you too, assuming your project settings match your video specs.
When you added your first clip to your timeline, did you see a yellow orange "render" line along the top of the timeline, above this first clip? If you do, then the program has not properly set up its setting to match your video specs and you will need to do so manually if you want the best quality and performance.
I do see that line, but don't see how I can adjust the settings. If I go to Edit ->Project Settings everything's grayed out. At least, everything that seems like it would be significant. I don't need to render the work area do I for better export quality? In fact I thought I read something saying not to do that. I at least feel like progress might be being made here, so thanks for the help. Attaching screen shots of my grayed out settings panels.
The only other thing I can think of is that two of the clips are at 60fps instead of 30 (ok, 29.97, etc.). Not sure why, apparently I captured them that way and now FCPX won't let me adjust those properties. The main movie is 29.97, and that's the one I'm really concerned about the quality.
P.S. Oh, and the exporting for dvd authoring in another program won't work, I'm trying Elements because none of the other DVD authoring programs on the Mac are working for me under Sierra. Seriously investigating rolling my iMac back to El Capitan where things still worked.
What model of camcorder is your video coming from and what is the resolution, frame rate and format of the video?