"What should White do about this?"
In the King's Gambit, the answer to this question is typically, "Attack!"
This is an offshoot of the Quaade, rather than the Quaade proper, coming from the Fischer defense (4. ... d6) move order. The idea behind ...d6 is to prevent Ne5. So White determines to just go ahead and ignore the threats and sacrifice a piece for even more impressive development. For example, after Qf3 above, White seems ready to play 0-0-0 and push e5 to blow open the center with Black's king stuck in it. Every one of White's remaining development moves comes with solid threats.
It's been a while since I've been in this section of the jungle, but as I recall White's threats are strong enough to recover the piece, coming out perhaps still a pawn down. But the game is complicated, and seems harder for Black to play than White, which means time to dust off the old saw, "good practical chances."
I think the main point in learning openings like the King's Gambit is to learn how to convert material into time, and back again. (A King's Gambiteer doesn't care overmuch for pawns or pieces when the King is in their sights.) Sacrificing the f3 Knight for development and open lines is a common theme; it occurs in the Rice and Muzio gambits as well. (True in the Rice the Knight is sacrificed on e5, not f3, but the idea is similar. Muzio has White playing Bc4 instead of Nc3, after which Black usually opts out of ...g4 by playing Bg7, giving you an idea of the nightmarish complications that would need to be solved over the board with a clock ticking.)
The Shaw book was always problematic to me. It's edited down from his Magnum Opus on the KG (IIRC, it was a huge multi-volume set) and walking the line between useful and insufficient while condensing that material down is a difficult task that I'm not sure was always successful. Since you're going through this as a learning exercise, I'm just going to suggest a few ideas for you to check out on your own; exploring all the resources in the position will help you to build your own understanding of what's going on and how it all fits together.
The heritage from the Fischer Defense suggests Ng1 as an alternative to Bxf4 above. The difference between that position and the mainline Fischer is White's Nc3 instead of h4, and that might prove to be an improvement for White over the "normal" Fischer lines. The main idea with that is White will take a step back and now try to exploit the weakness of Black's advanced pawns. The loss of a tempo is merely incidental, White still has a development advantage, since Black has wasted so much time moving pawns.
If you're feeling a mite more adventurous, you could substitute Bc4 for d4, then if Black would continue with the idea of ... g5, you could examine 0-0 (once again the Muzio can serve for inspiration). Doubling Q and R on a file facing two isolated pawns right next to the Black King (after ...gxf3 and Qxf3) is an idea worth exploring. These positions can get complex, but you need to overcome your reflex to move or protect attacked pieces, and realize your opponent is giving you two free moves (one by attacking the piece, and another by taking it) so look for how you can make the most of it. An axiom is three moves are worth a pawn; in that position White is up more moves than that.
Following Shaw's line (which could very well be best) my own first instinct in the position after Qxf3 would be to examine how quickly White could make the pawn break e5 operational. I think at a minimum 0-0-0 is required first, but that might be all. The idea of White blowing open the center while Black's King is still there (and, just for added spice, the Black queen is staring down at a protected White Rook sitting on d1) is captivating. After the break, stir in the light squared Bishop heading for either c4 or b5, and the other rook coming to e1 or f1, and things are getting a little warm on Black's side of the board. (Remember, the B on f4 can move, unveiling an attack from the White queen on f6/7. Paired with a B on c4 or a R on f1, that in itself could be worth a Queen or Rook in some lines.)
I'm not going to promise you a machine won't win in these lines as Black. Stockfish and LeelaZero played a couple of these lines if you get curious about that (Alexander Morozevich did as well). But the point is you're not playing machines; you're playing humans. If you don't end up playing the King's Gambit, the knowledge gained will still be useful. I think every improving player should spend some time learning about gambits; after all, you can't forestall threats if you don't understand how they are created. World Champion Tigran Petrosian, one of the most solid and subtle positional players we've ever seen, was also one of the best tacticians of his day.