I had been working on the question on and off for the past eight months.
I went to great lengths to answer this, even checking leads that had been largely dismissed. Apart from the obvious search attempts, here are some specific things I tried.
Consulting with R.L. Stine
The OP mentioned,
I don't think it was R.L. Stine, but I cannot remember the author.
I wanted to be absolutely sure, given the R.L.-Stine-esque nature of the description, and so in July of this year I contacted him directly:
@RL_Stine Quick question, if you have a moment. There is a now-famous question on a few forums regarding a horror story...
@RL_Stine According to the asker, the narrator is a boy who was scared only twice a day for exactly one minute, when the clock read 3:33.
@RL_Stine At those times, the 3's would be powerful enough to leave the clock and run around terrifying the boy!
@RL_Stine The asker suggested it might have been one of your stories. Ring any bells? Thanks for your time and have a nice day. :-)
@RemnantOfPraxis Not my story. Let me know if you find out whose.
I was impressed that Stine responded within an hour. At least that definitively ruled out a large chunk of horror literature. (Of course, I have updated Stine regarding the solution to the mystery.)
Visit to a horror specialist
In August of this year, I had a two-hour meeting with a librarian in the city I currently live in, who curates the library system's horror collection. He has an encyclopedic knowledge of 20th Century horror short stories, but couldn't think of a "3:33" short story that matched this one in both content and time period. The ones he brought up that were closest in content and time where ones that I had already ruled out.
Interestingly, he did ask me if it could have been a number other than 3, as there are plenty of horror stories involving clocks and various times. However, I mistakenly vetoed this, asserting that the OP must have gotten that main detail right!
To repent for my mistake, I, along with @MikeEdenfield, reached out to @Zompz (the OP, who hadn't been active on the site in about a year) to let her know that the mystery had been solved by our new expert sleuth, @KyleHale.