The Giant Cores are probs the biggest hold up in the game, they are fairly rare and only drop from those two enemies (probably more common via the skywatchers) I'd say just get the 6 cores from the shrine chests as those are enough to max the Ancient Armor.
There is sadly no fast way of...
Theres 6 Giant cores in various shrine chests, Skywatchers and Stalkers are the only ones who can drop Giant Cores you won't find them from scouts or trial guardians. Note it is stalkers not damaged Guardians that drop this.