Although "full of" is frequently used in english including America it is not good grammar.
If that is true, then the dictionary is full of errors :
Just click on those links and take a look at the examples.
See some examples below, copied from those pages:
1.2 (full of) Containing or holding much or many; having a large number of.
‘his diary is full of entries about her’
A2 containing a lot of things or people or a lot of something:
This sweater is full of holes.
His essay was full of spelling errors.
I'm full of admiration for you.
You're always so full of energy.
If you have a source where it says that it is not good grammar, could you please post that link here ?