"Then I went back into the checking account register and did a negative amount for the PayPal fees so the correct amount would be deposited."
That is one way to do it.
"I then deposited the fee amount back into my PayPal Account (Bank)."
That is the mistake. You never enter the same data twice, and this is not a deposit. If you already reduced your Real bank deposit by entering the fee as expense there, you stop there. Don't keep making entries.
"Now I am trying to now move that fee amount from PayPal Account (bank) to PayPal fees (expense) in QB."
Look at your expense account; you already did this based on your initial statement. That additional entry you made as a deposit to PP is to be deleted.
"Also going forward what is the best way to disperse the fees to the PayPal expense account, and the rest to the PayPal bank account?"
"PP" expense is simply Banking Expense.
In the PP Bank, you should have deposited your Customer's full payment here. Then, in the PP bank account you enter the fee you incurred.
Then, when money has been settled to your real bank account, this is not Sales or Income again. It is just a transfer, from PP to Checking.