The word 'promiscuously' is correct and usable in written English.
It is an adverb, usually used to describe someone or something behaving in a manner that is indiscriminate or not showing discrimination. For example, "The teenager acted promiscuously and went to parties with people she had just met."
Dictionary
promiscuously
adverb
In a promiscuous manner.
Exact(60)
THIS ill-tempered, promiscuously focused election should be about the economy and if the Conservatives pull off the win that their ratings on the issue seem to warrant, it may yet be.
But the main reason people want the black card is that it is so difficult to get.That elusiveness is unusual in South Korea, where credit cards are issued promiscuously.
By putting their faith in numbers, horses and playing cards, gamblers assert, in effect, that luck matters in life, and that net worth has little or nothing to do with moral worth.In the 18th century, respectable folk were shocked by the way people of all classes and colours mingled promiscuously at cockfights.
Teenagers watch less TV than any other age group, and when they do turn to television, it is to choose promiscuously from among 100 channels: most of them have never known anything but cable.
This is in large part due to the current unworkable mix of virulently partisan political views with promiscuously opportunistic horse-trading.
Parks such as St James's, which are eulogised for their orderly planning in one poem ("In such green palaces the first kings reigned"), become very different places at night in another ("Footmen, fine fops, do here arrive,/And here promiscuously they swive").
It was a process in which history, geography, religion and myth were promiscuously pressed into service.
But even if it was not, that would not have been significant, because at this early time matter could change promiscuously into antimatter, and vice versa, at a rate dependent on how much of the other sort was around.
In the case of bacteria, which seem to exchange genes promiscuously, that seems an excellent way of looking at things.In a sense, such profligacy extends to humanity, too.
IN AN age where people promiscuously post personal data on the web and regularly click "I agree" to reams of legalese they have never read, news of yet another electronic consent form might seem like a big yawn.
When the English later themselves became conquerors, they promiscuously took on words from languages all around the world.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com