New Bern is the larger city with much more preserved history, so I'd have to give the nod to New Bern. Both are equally safe, which is to say, as a visitor in a hotel following normal cautions, you don't have anything to worry about. We're not talking thugdoms!
Washington is a small town of about 10,000 people. Most of its older buildings were destroyed in the numerous civil war shenanigans, and it never had a pedigree as impressive as New Bern's. It is developing its waterfront as a tourist destination, though, and if you were spending time in the area, I would recommend it as a nice town for an afternoon visit. Think Beaufort, but on a much smaller scale.
Enjoy!
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