My opponent argued that because this was my “final” play (last card in my original hand), I was done playing all, and would not be able to play all of the new cards from the hand I’d just traded for.

A: You’ll be happy to know that you were correct, not your opponent. After the Trade Hands is completed, you now have cards in your hand, and the Play All rule says you keep playing as long as you have cards in your hand, so your turn doesn’t end, you just keep going.