@Eve2019 

Just to make sure we're using the same terminology, let's refer to your vendor's shipping cost that your vendor bills to you as Freight In (FI), and the cost that you incur to ship product to your customer as Freight Out (FO).

Th IRS does not require you to capitalize (add to the value of your inventory) the cost of FI if your business qualifies as a small business (less than $25M annual sales).  Therefore, you can expense it immediately.  It's up to you whether you want to record it as a COGS or an expense.  Both have the same impact on net income so it depends on where you want to see this listed on your P&L.  Personally, I think it makes the most sense for FI to be a sub-category of COGS. 

FO is an expense, not COGS.  Also, packaging supplies are an expense, not COGS, as you mentioned.