Here is the rule:
When the ball becomes dead in a side zone, is unfit for play, is subject to
measurement in a side zone or is inaccessible, a replacement ball shall be
obtained from a ball person (A.R. 1-3-2-I).
The referee or umpire shall determine the legality of each ball before it is
put in play.
If we have a run to the opposite sideline, we are not going to get a new ball from clear across the other side of the field, unless there is something wrong with the one we have. The Referee and Umpire, by rule determine that, not the teams. Here is the play situation noted in the rule above:
AR. 1-3-2-I: On fourth down, kicker A1 enters the field with an approved game ball
and asks the referee to substitute it for the ball used during the previous
down. RULING: Substitution of the ball is not permitted.
