Yes, Dumbledore asked Snape to kill him in order to preserve Malfoy.

As seen through Snape's memories in Chapter 33 of Deathly Hallows:

Ultimately, of course, there is only one thing to be done if we are to save him from Lord Voldemort's wrath."

Snape raised his eyebrows and his tone was sardonic as he asked, "Are you intending to let him kill you?"

"Certainly not. *You must kill me."

And a bit later:

"If you don't mind dying," said Snape roughly, "why not let Draco do it?"

"That boy's soul is not yet so damaged," said Dumbledore. "I would not have it ripped apart on my account.

Thus, we see that Dumbledore was worried that Voldemort would kill Malfoy, if Dumbledore was not killed, but Dumbledore also wanted to protect Malfoy's soul. He was able to save both Malfoy's life and soul by having Snape kill him instead.

Yes, Dumbledore did ask Snape to kill him so Draco wouldn’t.

Dumbledore did indeed ask Snape to kill him, so that Draco wouldn’t either succeed and maim his own soul, or more likely fail and be killed by the Dark Lord for his almost inevitable failure.

“All the same, try. I am concerned less for myself than for accidental victims of whatever schemes might occur to the boy. Ultimately, of course, there is only one thing to be done if we are to save him from Lord Voldemort’s wrath.’

Snape raised his eyebrows and his tone was sardonic as he asked, ‘Are you intending to let him kill you?’

‘Certainly not. You must kill me.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

Dumbledore didn’t want Draco to do it and harm his soul, but he also didn’t want to die at the hands of someone who would prolong his death and make it more painful. He preferred that Snape kill him quickly, and since it was done as an act of mercy, Snape’s soul would remain intact.

“If you don’t mind dying,’ said Snape roughly, ‘why not let Draco do it?’

‘That boy’s soul is not yet so damaged,’ said Dumbledore. ‘I would not have it ripped apart on my account.’

‘And my soul, Dumbledore? Mine?’

‘You alone know whether it will harm your soul to help an old man avoid pain and humiliation,’ said Dumbledore. ‘I ask this one, great favour of you, Severus, because death is coming for me as surely as the Chudley Cannons will finish bottom of this year’s league. I confess I should prefer a quick, painless exit to the protracted and messy affair it will be if, for instance, Greyback is involved – I hear Voldemort has recruited him? Or dear Bellatrix, who likes to play with her food before she eats it.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

It’s clear that Dumbledore did both ask and expect Snape to kill him, and this was always his plan.

“After you have killed me, Severus –’

‘You refuse to tell me everything, yet you expect that small service of me!’ snarled Snape, and real anger flared in the thin face now.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

So yes, Dumbledore did indeed ask Snape to kill him to spare Draco the Dark Lord’s wrath or a maimed soul because of Draco’s own attempts to kill Dumbledore.

But it wasn’t he thought the Dark Lord’d kill Draco for the wand.

However, though Dumbledore did ask Snape to kill him, it wasn’t because he thought that the Dark Lord would kill Draco for the Elder Wand. Dumbledore had intended Snape would get it.

“He believes that the Elder Wand removes his last weakness and makes him truly invincible. Poor Severus …’

‘If you planned your death with Snape, you meant him to end up with the Elder Wand, didn’t you?’

‘I admit that was my intention,’ said Dumbledore, ‘but it did not work as I intended, did it?’

‘No,’ said Harry. ‘That bit didn’t work out.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)}}

He likely didn’t expect that Draco would disarm him, so he didn’t think Draco would be in danger from being thought the master of the Elder Wand - he thought Snape would have it.