So, I should start out by saying that this definitely isn't a complete answer- this is just my own experience coupled with my own research, and I don't mean for this to be a definitive answer.
I'm a teenager from Wisconsin, and during my childhood, I, too, was taught that certain words were "bad" words. And unlike in your experience, we did call them 'swears,' at least when we learned them. By the time we learned what they actually referred to, we had long since outgrown them. Today, I associate 'a swear' as going hand in hand with 'My mommy says' and 'I'm telling!'. It's definitely rather puerile.
However, it's certainly not unheard of; just as an example, Oxford Languages has it as a noun:
a word regarded as coarse, blasphemous, or otherwise unacceptable in polite or formal speech; a swear word.
It's also useful to look at Wiktionary's etymology for the word 'swear':
The original sense in all Germanic languages is “to take an oath”. The sense “to use bad language” developed in Middle English and is based on the Christian prohibition against swearing in general (cf. Matthew 5:33-37) and invoking God’s name in particular (i.e. frequent swearing was considered similar to the use of obscene words).
So it would make sense that 'swear word' came first, as it was a word invoked while swearing. The earliest entry I can find in a dictionary for 'swear word' is from 1891:
swear-word (swár' werd), n. A profane word; an oath. [Colloq.]
There has been in the post an immense quantity of scolding, occasionally a swear word.
Elect. Review (Amer.), XIL I. II.
Though it's definitely from much earlier. Like here, from 1804:
Members of tenants connected with my firm where, if a man used a swear word upon their premises, he would not be allowed to drink and not be allowed to enter the premises for a month.
(Etymonline falsely states that it's from 1873; there's also another snippet from 1836.)
Here are a few early examples of 'swear' as a noun:
Papa said a swear, unbuckled his sword, and then got out to crank.
(1906)
I suggested we do that and Jim was nearly cross. "Look at the gauge," he muttered with his head upside down behind the radiator.
I did and it said eight gallons, but Ned already had the cushion out and the tank-valve open. There was scarcely a smear of gas in it. Jim came to look and said a swear.
"The blank, blank gauge is on the blink!" he growled and gave it a vicious poke.
(1915)
And for some early examples of 'swear word,' see Nimpo's Troubles (1880):
As soon as Anna reached home, she went to her mother. "Mother," said she, "Nimpo Rievor swears!"
"What!" exclaimed Mrs. Morris, looking up.
“Nimpo said a swear word to me to-day," Anna replied.
"What did she say?" asked Mrs. Morris. Anna repeated what she had understood Nimpo to say, and her mother was much shocked.
"Don't say it again, Anna," said she severely, "even to whisper it. And I forbid you to play with her any more. I am afraid she is getting into bad ways since her mother went away. It ain't good for young people to be left to their own devices."
(Here's the earliest I can find- 1872- I excluded it because 'Nimpo' catches the imagination really quite greatly, though 'Izzy Paul,' 'Maria,' 'Flyaway,' and 'Mrs. Morris' also come rather close.)
My guess is that since it's colloquial without being bizarre/non sequitur, 'swear' as a noun won't be too widely documented or studied.
So to go back to your questions:
Is my variety of English unique in that I use "swear word" as the standard term and only use "swear" as a verb? Is this difference divided regionally, and if so, where is each usage preferred?
No, it's not unique, and I'm quite certain that your variety is more standard, just based on the fact that 'swear word' is far more common than 'a swear,' and seems to predate it.
And as for the regional division: To be quite honest (and this is more speculation than certainty), I'm going to say that there is no regional division; American parents have quite a surplus of terms to use when referring to such words, and 'swear' just happens to be one of them. The fact that it's so often used by parents explaining stuff to kids colors it as being childish, and this means that it isn't going to be used and picked up by the type of people who popularize words. I almost exclusively use it to make fun of my friend's little brother (don't worry, he's horrid, I don't need to feel guilty). So little kids learn that 'a swear' is a bad word, then they grow up and dismiss 'a swear' as childish, and they start calling them by whatever the term in vogue is instead.