Senator Joseph R. Biden Jr., declaring that his candidacy had been overwhelmed by ''the exaggerated shadow'' of his mistakes, dropped out of the race for the Democratic Presidential nomination today. The 44-year-old legislator from Delaware betrayed little bitterness but some sadness as he stood behind a forest of microphones at a crowded news conference. In sharp contrast with former Senator Gary Hart, who withdrew from the Democratic contest in a blur of invective against the news media, Mr. Biden chose to swallow most of his private anger and place the burden for the end of his candidacy on himself. ''I...