A Google search put a British Common Sense Party at the top of the list.
As far as the U.S. goes, almost every link I found did not find a current Common Sense Party.
https://ballotpedia.org/Common_Sense_Party had a rather hard-nosed idea of common sense, sort of casting blame all over the map. The “Official website” link gave “Server not found”.
http://www.commonsenseparty.org/home.html has lots of common sense quotes, but the website is for sale. It is copyright by the Common Sense Party who will consider all serious offers. It can be reached through its contact page: http://www.commonsenseparty.org/contact-us.html
The Centrist Party seems like a believable alternative. Its website is http://uscentrist.org/ which is currently active. The latest blog entry is dated 2017-03-28: http://uscentrist.org/news/the-common-sense-blog-the-ins-and-outs-of-congress
Its philosophy does use unclear definitions of “free market, limited government and individual liberty” but does call for “protecting the common good”. We should consider that the balance between the two goals will always be unclear.
A Stanford student, Kyle D’Souza, gave a reasonable call, “The common sense party” http://www.stanforddaily.com/2017/01/13/the-common-sense-party/. This was in January. It only had one comment which I thought was off the wall.
The American Common Sense Party seems to be a one-person party on Facebook, the last entry being 2016-03-09.
The Common Sense Party also is on Facebook. It’s latest entry, 2017-03-22, was a link to Washington Post article of the same date: “Ex-Colo. GOP leader said only Democrats committed voter fraud. Now he’s charged with voter fraud.”
There are other recent articles about a “Common Sense Party”, but my quick scan showed that some are only local, not national or even state-wide.
Like the then moderate Republican Party arose from the Whig Party, maybe a true moderate party will arise from the current non-Lincoln Republican Party.