Joseph R. Biden Jr. came up in politics as an old-school backslapper whose greatest strength was his ability to connect. He doled out handshakes and hugs to friends and strangers alike, and his tendency to lavish his affections on women and girls was so central to his persona that it became fodder for late-night television jokes.

But the political ground has shifted under Mr. Biden, and his tactile style of retail politicking is no longer a laughing matter in the era of #MeToo. Now, as he considers a run for president, Mr. Biden is struggling to prevent a strength from turning into a crippling liability; on Tuesday alone, two more women told The New York Times that the former vice president’s touches made them uncomfortable.

For Mr. Biden, 76, the risks are obvious: the accusations feed into a narrative that he is a relic of the past, unsuited to represent his party in the modern era, against an incumbent president whose treatment of women should be a central line of attack. Mr. Biden has denied acting inappropriately but has said he will “listen respectfully.”