[jack09268]

Even though the vowel "a" is dropped from the spelling, if you actually say it out loud, you do pronounce a vowel sound when you get to that spot in the word, something like "shouldn'tuv", whereas the "o" in "not" is dropped from both the spelling and the pronounciation.

[SAI_Peregrinus]

The pronounced vowel is different than the 'a' in 'have'. And the "h" is definitely elided.

[int_19h]

Many English dialects elide "h" at the beginning even when nothing is contracted. The pronounced vowel is different mostly because it's unstressed, and unstressed vowels in English generally centralize to schwa or nearly so.