"I've" is not normally used where it is spoken as "I have". This is similar to people using "an" in front of a word starting with "h" where the "h" is pronounced. Not sure why people do this.
I think the distinction sidegrid was making is not actually about pronunciation but about the two uses of have as an auxiliary verb and as a verb indicating possession.
My impression is that standard American English only contracts the auxiliary verb ("I've biked up Mount Tamalpais") and not the possession verb (?"I've a pair of prescription sunglasses"). Hence "I've been diagnosed with bronchitis" (auxiliary) but not ?"I've a case of bronchitis" (possession).
However, I think this rule is different in Commonwealth English, so we might just be witnessing a difference in English varieties.
As a Commonwealth English speaker, I am not aware of a rule preventing contraction of the possessive verb "to have." Maybe there is, and I'd be interested to learn about it.
Having said that, commonly I wouldn't say, "I've a case of bronchitis," rather just "I've bronchitis." It feels more natural to say "I have a case of bronchitis." However, this may just be personal preference.
Are you saying people don't use "I've" in spoken language? Because it see it regularly, "I've never" is even a drinking game.