Coast Guard leaders have alerted veterans that if political leaders don’t resolve the partial government shutdown in the next few weeks, retirees’ families won’t get the pension payments scheduled for delivery on Feb. 1.

Unlike the military branches covered by the fully funded Department of Defense, the Coast Guard and parent agency the Department of Homeland Security are furloughing staff while President Donald Trump and congressional Democrats feud over funding for his border wall.

With the record-breaking standoff stretching past its third week, Adm. Charles W. Ray, the Coast Guard’s vice commandant, recently broke the news to his people that...