Band Played 'Dixie'

Then, as the rain fell and a country-and-western band played ''Dixie,'' a tune avoided in Mr. Carter's campaigns, Jimmy and Rosalynn Carter took each other in their arms and danced a kind of polka across a makeshift stage that had been erected in the town's block-long Main Street.

The former President seemed relieved and genuinely happy to be home. The last 48 hours had been an agonizing vigil in the Oval Office as he and his aides waited for word that the hostages were on their way. That message finally reached him about 12:30 P.M. while he and Walter F. Mondale were driving to Andrews Air Force Base after the inauguration.

Later, on the flight home, Mr. Carter received a call from Deputy Secretary of State Warren Christopher, advising him, as he said later, ''officially and for the first time that the aircraft carrying the 52 hostages had cleared Iranian air space and that everyone of them was alive, was well and was free.''

The call was the climax to the final 48 hours of the Carter Presidency, a struggle against the clock to win the freedom of the hostages before his term in office ended. The hostage crisis dominated Mr. Carter's final two days as President just as it did the last 14 months. Germany Trip Delayed

Arriving here in midafternoon, Mr. Carter and his staff decided to postpone until tomorrow the trip he had planned to take tonight to West Germany to greet the hostages. The former President was exhausted, the staff explained, and he wanted to give the former captives a few hours to acclimate themselves to their new surroundings in Wiesbaden, West Germany.