In the rulebook, it says that a player must resolve as much of an action as possible. So would that mean that even though I can't resolve the trashing part, I still resolve the gaining a card part?
Exactly. Always do as much of the card as you can. Since with Feast gaining a card is not contingent on trashing the card, you get to gain a card even if the card was already trashed with the first Throne Room. Other cards have "If you do..." language that interact differently with Throne Room, so keep an eye out for them.