Is it true that the Covid vaccine causes a cough a month or 2 after getting it?

My dad said he’s been hearing stuff on the radio about fully vaccinated people dying from Covid or people developing a cough a month or so after it. And for the past couple weeks I’ve been having a bad cough that’s been getting worse, and it even makes me gag and throw up. I have coughing fits where I feel like I can’t breathe and I get so out of breath and my body temperature feels like it goes up with it. I can’t do anything like laugh or eat without having to worry about coughing, gagging and throwing up and at night I’ve been coughing like crazy and it’s a dry cough. And in the mornings I feel like I have to throw up constantly and it’s hard to stop even though I get so out of breath and hot, and nothing comes out except pinkish stuff cuz of an empty stomach. And I’m fully vaccinated. I got the first dose of the moderna in mid May and the second dose in mid June. I just want this to be over already.
Is it true that the Covid vaccine causes a cough a month or 2 after getting it?
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