"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.
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.
https://en.wiktionary.org/wiki/have#Verb
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.
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.