When did it become fashionable to drop t's in certain words?

Sometime in the 1500s.

"How to Pronounce 'Often'"

There’s a t in often, but how often do you hear it? As you might guess, the t was pronounced in the past, when the word began as a variant of oft (also spelled ofte in Middle English), which was the more common form until the 1500s. Oft is now archaic for most of the senses of often, but is still used in compound adjectives like oft-repeated and oft-quoted. Ofttimes and oftentimes both carry that archaic flavor but are still in active use. After the -en suffix was added to ¬oft, the t fell away in pronunciation, but remained in the spelling.

Similarly, the medial t in words like soften, hasten, and fasten was originally pronounced, as the -en was added to base words that were recognizable (soft, haste, fast). Listen is a bit different; although the archaic verb list exists, listen comes from the Middle English listnen, and evidence is that t after s and before n was not pronounced.

https://www.merriam-webster.com/words-at-play/correct-pronunciation-of-often

But this is probably a different phenomenon than the t-glottalization you're noticing. According to the Wikipedia article, "The earliest mentions of the process are in Scotland during the 19th century" ("T-glottalization"). Most of the written materials I can find focus on British accents, but this paper about t-glottalization in American accents is interesting: "T-GLOTTALIZATION IN AMERICAN ENGLISH"

The authors cite a 2006 study that suggests that t-glottalization is most common among adolescents in America. They also suggest that it's more common on the West Coast, then the East.

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Anecdotal

My own experience, having grown up on the West Coast and moving to the East Coast as an adult, actually suggests the opposite.

Actually, the t-glottalization in the Northeast may not be more frequent, but it seems somehow "stronger." That is, I and virtually everyone I know (including my parents, who are in their sixties) pronounce button with a glottal stop, not a /t/. However, I have met a number of twenty-somethings raised in the Northeast who pronounce button differently, as if their tongue tip doesn't touch the roof of the mouth until the n. I believe that when I say "button," my tongue tip rises to make an unreleased t, perhaps simultaneously with the glottal stop.

I have only noticed it also in the last 4-5 years more often in the word "important" pronounced [impor 'ehnt].

This is different than the 't' being pronounced as a 'd' [impor 'dant] which seems to be more a regional thing that I have heard in other areas when I travel.