Although this may put the definition of "professional" in question, there exists a North American club team that's existence does date before the Cincinnati Red Stockings.
Some lesser known history, there exists a sport in North America that has been around for many centuries. That sport is lacrosse.
Believed to have originally been played by the Native American Iroquois people, historians have traced the general sport of lacrosse back to the 1100's.1 The game was predominately played by those Native American people in what would become the North-Eastern United States, and lower parts of Ontario and Quebec. Explorers and missionaries have recorded details about the game as early as 1637.2
As time wore on, the people of Canada grew fond of the game and began to play it themselves. According to the "Canadian Encyclopedia":
"In the 1830s, visiting anglophones from Montréal noticed the games and
learned to play from their Mohawk neighbours, adopting the French term
lacrosse for their new pastime"
They enjoyed it so much that a few decades later in 1856, enthusiasts banded together and formed Canada's first ever lacrosse club the "Montreal Lacrosse Club." 3
Now, it was around this time that the Prince of Wales traveled to Canada and grasped the merit of holding sporting exchanges between Britain and Canada. As a result, tours began developing in Britain pitting the Native American Khanawake players from Montreal against white gentleman amateurs from Scotland, Ireland, and England.4 It was then, in 1867, that one of the first ever "professional" sports teams/ clubs was formed. The Montreal Lacrosse club, then led by team captain W.B Johnson, began organizing matches in the British nations primarily as a profit making enterprise. The team was comprised of 16 Khanawake players who were each paid $25 dollars for their play (roughly $1600 dollars today).5
Given OP's qualification of "professional" as " (a team) that paid its players as professionals." These Khanawake players would've fit the category. As for the club still being in existence it is complicated. The club itself is not in operation like the Boston Red Socks are, but it exists in an almost ceremonial way. This article outlines the 150 year anniversary of the Montreal Lacrosse Club's existence. The anniversary was celebrated by an exhibition game between the sports two oldest clubs, one of which being Montreal Lacrosse Club. So yes the club is still in existence, but does not operate as team which is comprised of players who are paid a salary by it.
Sources:
Liss, Howard (1970). Lacrosse. Funk & Wagnalls. p. 13.
"John De Brbeuf.” Patron Saints Index: John De Brbeuf,
“Montreal Lacrosse Club.” Wikipedia, Wikimedia Foundation, 25 May, 2017
King, C. Richard. “Lacrosse Exhibition Tours.” Native Americans in Sports, Taylor and Francis, 2015, p. 178.
King, C. Richard. “Lacrosse Exhibition Tours.” Native Americans in Sports, Taylor and Francis, 2015, p. 178.