A proposed shift by the Boy Scouts of America to drop its national ban on gay leaders and scouts, and allow local scout units to decide for themselves, was the center of attention as the organization’s national board gathered here on Monday for a three-day meeting and a vote on the issue. But the undercurrents of the debate — a drop in participation in the Scouts over the last decade and a deep division between conservative and liberal church groups over the proposal — are raising the stakes even higher for the vote as a kind of proxy on the question of how scouting stays relevant in a changing social climate, Scout volunteers involved in the discussions said. The strains on the historic youth organization were evident, and visible. At the organization’s national headquarters, also here in Irving, supporters of eliminating the ban dropped off petitions that they said had been signed by 1.4 million people. During the weekend, President Obama, in an interview, said he favored allowing gay youths to join the Scouts, while Gov. Rick Perry of Texas expressed the opposite opinion.