I think there are a couple related but slightly distinct things that you're talking about.
At the end of the word Connecticut, you are probably talking about replacement of /t/ at the end of a word with a glottal stop [ʔ]. This also can affect /t/ before a consonant in the middle of a word, as in wetly.
In words like important, the relevant sequence is even more specific: you're talking about the pronunciation of /tən/, a /t/ followed by the unstressed vowel phoneme called "schwa" and then the nasal consonant /n/. In many accents of English, the phoneme sequence /ən/ can be pronounced as a syllabic nasal, transcribed [n̩], in certain contexts, including after /t/. The realization of the /t/ itself is a little variable: it can be a nasally released /t/ sound (transcribed [tⁿ]), or for some speakers it can be a glottal stop [ʔ].
As far as I know, the use of [ʔn̩] is fairly widespread, so it wouldn't stick out too much. A slightly different thing that I think you might be noticing is the use of oral release in this context. I said earlier that /tən/ is often pronounced with [n̩], but some speakers do in fact use something like [ʔən], with a glottal stop followed by an oral vowel followed by a nasal consonant.
Speakers who don't usually use oral release seem to find that the pronunciation of /tən/ with oral release sounds odd, but they describe it in different ways. I've seen some people say it sounds to them like the insertion of an /ɛ/-like vowel; another common perception seems to be loss of the /t/ sound. A recent Reddit thread on this subject: https://www.reddit.com/r/linguistics/comments/fq3vfz/millennialzoomer_glottalization/
A post there by problemwithurstudy links to the paper "Where are the mountains in Utah?", by David Eddington, which studies the topic of how /tən/ words are pronounced in Utah. Eddington found that what is popularly characterized as "dropping t's" in Utahn pronunciation is actually the use of a glottal stop with oral release. I don't know what the situation is in Connecticut but I wonder if it could be similar.
Decapitated Soul's comment "'Mountain' pronounced with a 'glottal stop' sounds awkward" reminded me that /ntən/ can behave differently from /tən/ for some speakers, but I don't know the details. Here's a 2016 Language Log post by Mark Liberman about the pronunciation of the name "Clinton": https://languagelog.ldc.upenn.edu/nll/?p=27112