lord of the rings - Would the dead answer to Aragorn if he DID NOT have Andúril?

The Dead follow Aragorn because of who he is, not the sword he carries.

In the books there's none of this "sword-fighting with the Dead" nonsense. Aragorn doesn't even get his sword out; it's enough for him to proclaim who he is and show his banner, and the Dead follow him without question.

To that Stone the Company came and halted in the dead of night. Then
Elrohir gave to Aragorn a silver horn, and he blew upon it and it seemed to
those that stood near that they heard a sound of answering horns, as if it
was an echo in deep caves far away. No other sound they heard, and yet they
were aware of a great host gathered all about the hill on which they stood;
and a chill wind like the breath of ghosts came down from the mountains. But
Aragorn dismounted, and standing by the Stone he cried in a great voice:

'Oathbreakers, why have ye come?'

And a voice was heard out of the night that answered him, as if from
far away:

'To fulfil our oath and have peace.'

Then Aragorn said: 'The hour is come at last. Now I go to Pelargir upon
Anduin, and ye shall come after me. And when all this land is clean of the
servants of Sauron, I will hold the oath fulfilled, and ye shall have peace
and depart for ever. For I am Elessar, Isildur's heir of Gondor.'

And with that he bade Halbarad unfurl the great standard which he had
brought; and behold! it was black, and if there was any device upon it, it
was hidden in the darkness. Then there was silence, and not a whisper nor a
sigh was heard again all the long night. The Company camped beside the
Stone, but they slept little, because of the dread of the Shadows that hedged them round.

_{-- The Return of the King, Chapter 2: The Passing of the Grey Company (emphasis mine)}