I understand this problem, pretty much routine with many first time parents, and mroe so when the child has become ill during the first few days of life
However, while this may sound difficult, the only solution to this is to worry less :)
You need to know a few danger signs, like excessive irritability, complete lack of feeding, limpness, difficulty breathing, high fever (gen above 102 F) and an unwell looking child.
However, if the child is otherwise active, playing, smiling and apparently normal
then a little bit of sneezing & coughing is absolutely normal
and should not construe anything serious
Talk to your pediatrician regarding your concern, and you may also take the help of the nursing staff & an elder relative/ friend.
I know that sometimes small babies can become overwhelming,
but almost all of them turn out to be fine
Try to take some time out for a few minutes every day, wherein you can tune out and take a break and get some relaxation.
Ask your partner/ friends to chip in for a small period of time
And relax so that this time remains special ....
All the best