I can understand your concerns.
Better have her seen by your ENT doc who has done the surgery.
I guess she will be on antibiotics in the post surgery period. And she will be having difficulty in swallowing following the surgery.
she has now will add much more to her difficulty.
Rather than self medicating with over the counter medications, its better if you have her examined and get the medications from your doctor.
But you need not worry that the mucus will get down to her lungs because tonsils and adenoids
are removed. Its not so. Cough itself is a protective phenomenon that propels out the unnecessary materials out of entry into lungs.
Do see your doctor.
Wish a speedy recovery for your kid.