Yes, indeed the fullback was a ball carrier- often the major one- back in the day. There were three backs behind the Quarterback who was "under the Center" in the T formation. They crossed the T which is where it is got its name. The guy in the middle might be a half to a full step deeper than the other two which is why he was called, Fullback; the others were called, Halfbacks. In those days, Mel Triplett was the fullback, Gifford and Webster, the halfbacks. Chuckin' Charlie Connery was the QB, the Giants first ever T formation Quarterback. But even before then Connerly's FBs included Gene "Choo Choo" Roberts and Eddie Price with Joe Scott, and Skippy Minisi, and Kyle Rote as the Halfbacks.
Before the Charlie Connerly era began, the Giants played the Steve Owen's "A formation" which was a variation the single wing. Paul Governali was the Wingback; Ken Strong was the Fullback, Joe Sulaitis the Blocking Back, and Howie Livingston was Tailback.