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I do remember this level quite well, it's one of the original levels which has not been interfered with. You need to work at removing all the blockers as usual, which you have done, but at the same time watch out for where the conveyor belt takes the dragons and where the egg exits are. You can see this very clearly on your screenshot. Most of the exits are at the top, so if you have enough moves left then you can try to not clear any candies below the dragons until they get up there. There are two exits on the bottom row though, so in your case wait until the dragon has moved up the conveyor belt above that exit. Hopefully you can then clear the two candies below it. If you can't, let it move on a bit further. If it drops onto the bottom row to the right of the exit it will drop in when the conveyor belt moves, as it moves from right to left in the bottom row.
Actually, as you have 9 moves left on your game, the dragon should get to the first exit on the middle row just in time if you do nothing (i.e. make matches on the top two rows that don't interfere with the rows below)