Check out Murphy's on 23rd and Strand. I was in their last week. They have done a lot of work since it was Connely's. You might be able to do both there. I was only in there for a beer and didn't get to check the whole place out.
Sonny's bar definitely has shuffle board. It's been a long time since my last visit so I can't remember about darts.
Molly's pub has darts for sure.