Singing, then Cough! sounds normal to me
Singing, then Cough! that sounds about right.
That is common, espcially if you are just starting out singing.
I always have sugar free Ricola cough drops and will take a break and try to soothe the voice with that. You may want to take it easy for a while if it starts getting rough and quit for the night. Then the next morning I would expect the coughing to happen again, and more sugar free cough drops to the rescue for that as well.
After a month or so of singing every day, it should get to be less and less frequent.
I recommend sugar free cough drops because many years ago I ended up having tooth decay, due to the sugary cough drops.
I was singing every day, and taking the cough drops every day.
Still have the cough drops ready when necessary.