How do I adjust inventory in QuickBooks

 I have been entering my bills in QB and applying the inventory to COGS-COGS account. ... At no point does the P&L or Balance sheet note the value of the inventory assets that were not sold and remain on hand to be sold.  ...  Also, throughout the month I have to write off inventories due to damage or shrink.

Then you are expensing the full amount of the purchase and there will not be an inventory asset value on the balance sheet.
And since you are expensing the purchase, there is nothing to adjust either.

Since you are not using QB inventory, you must use the periodic inventory method.  There are two ways to do periodic inventory, choose one and stick with it, you can not mix and match

1. Create an asset account called purchases and post all purchases of item for resale to that account.  Periodically, weekly, monthly, etc value the inventory on hand, subtract that value from the amount shown in the purchases account and do a journal entry for the answer to the subtraction
debit COGS for that value
credit purchases for that value

OR

2.  Post all purchases to COGS.  Periodically, but at least at the end of the year, you value the inventory on hand and do a journal entry.
debit the asset purchases account for that value
credit COGS for that value

Print the P&L
then reverse the journal entry
debit COGS for that same value
credit the asset purchases account for that value 

This last journal entry, moves the value of what was on hand at the end of year back to COGS so the cost will be counted against the new year sales.