Cold sores and babies please help

My baby is 4 months old. We have all been under the weather thfor last few days, including baby: coughing, sneezing, snotty nose. But he’s not seriously unwell. I noticed yesterday he has a small rash on his face. I’ve come up with a cold sore today and I’m so so scared that he has caught it and this is why he’s been sick. Will he be okay? What should I do. I’m terrifiedot will make him very very poorly.

He’s better today and seems better in himself. Not so snotty and unsettled.

My partner doesn’t believe that cold sores can make babies sick and won’t support me in wanting to take him to hospital or am I being dramatic? He’s not sick enough for hospital but now I can’t stop worrying about it.