I don’t live near Dover or Concord so can’t help there but I moved to NH from Providence. Providence is lovely and checks most of your boxes, specifically the downtown/night life, gluten free restaurants, healthcare and probably dating options.

Lots of great dog trainers in the Providence area too, I’d check out the r/Providence sub for that.

There are some scenic walking trails within 20 min outside Providence but everything is flat and no comparison to what you’d find in NH.