DFDS do Harwich - Esbjerg. I took it last year on route to Poland - it was a superb over-night crossing and the food was as good as any I've eaten anywhere. They definately get my vote.
According to THIS SITE there is no direct ferry route from Newcastle - Esbjerg.
You could always take the DFDS Newcastle - Ijmuiden boat. GPS fastest route says 499 miles and 8 hours riding to Odense in Denmark. DFDS arrives about 0930 so you could be in Odense by the evening. Beats the slog to Harwich.
DFDS also run from Immingham to Esbjerg. I used to use it when I drove trucks and seem to remember that they took tourist traffic too. Worth checking maybe.