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You cannot put 1920x1080 on DVD, only on Bluray. Please confirm that you are making a DVD, not a bluray.
You may get good results encoding in Encore (which can use adobe media encoder), but there are some variables you may not be able to control. The general view is that encoding outside of Encore, and indeed, using third party options, may do a better job.
If you are new at this just use Dynamic Link, set it to dvd and let Encore do the rest.
Encore CS6 does a pretty good job in making dvd's.