Any could be correct depending on when they occurred (past, present, future). For example: 1. He got a fever (past tense) 2. He had a fever (past tense) Just another way to say the same thing. 3. He has a fever (present tense) 4. He has got a fever (present tense) Still correct, just a little less common of a way to say it. 4. He will get a fever (future tense) The same applies for the word "cold". Just switch it out with the word fever above. Now, a cold and fever are two different things. You can have them at the same time, but they do not mean the same thing. - A cold is a common viral infection of the nose and throat.Usually causes a runny nose and sore throat. - A fever is a temporary increase in body temperature of 98.6°F (37°C). It usually causes muscle aches, body chills, etc.

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“Getting cold” means the temperature is going down, or it feels like the temperature is going down. “Your dinner is getting cold” “I’m getting cold. Let’s go inside.” “Getting a cold” means getting sick. It’s slightly better to say “catching a cold”. “Get” isn’t wrong, but its use when there is a common better word sounds a little sloppy.