Sometimes we shorten "shouldn't have" to "shouldn't've" or "shouldn't a" especially in casual speech - not so much in writing.

A few points. The full phonetic transcription should be /ˈaɪ ˈʃʊdnt həv ˈiːtn/. Even with shorten vowels, in most standard dialects, the fleece vowel /iː/ would not be reduced to the happy vowel /i/ As pointed out by Dan, the vowels of function words in English are often reduced to a schwa /ə/. However, in rapid speech, while the /həv/ would be likely be reduced to /ə/, it would all but be elided (omitted) due to the longer /iː/ which follows.

This isn't unique to English. Native speakers of every language find ways to use fewer sounds to make it easier to say frequently used words and phrases. Do a web search for "reduced speech."

Without a recording it's hard to tell, but a native speaker pronounces this like "shouldn't've." You will sometimes see it actually spelled that way in print. The "h" in "have" is silent, the "a" is a schwa (ə), and the "v" is almost unvoiced (because it's hard to switch quickly from the unvoiced "t.") And it's very soft. A lot of times the _'ve_ or "of" is spoken like a schwa. In quoted dialog, it is sometimes written out as just "a:" "I shouldn't a done it." Other times, because of the "v" being almost unvoiced, it is written as "of," "I shouldn't of eaten it." So "shouldn't have eaten" becomes "shouldn't've eaten" becomes "shouldn't-a eaten" but the vowel sounds of the "a" and the "e" of "eaten" join and blend. If you listen carefully there is probably a _faint trace_ of something in there between ˈʃʊdənt and ˈitən, but it might just be a tiny little schwa-like sound. And then, of course, native speakers fill in missing words from context. If I carefully said "I shouldn't eaten it," many native speakers might believe they had HEARD "I shoudn't've eaten it."

Difficult to say, but you are missing the /v/ in the middle. Maybe they are talking quickly.