The PDF was already mentioned.
For the general plot problem one thing you should do is to shift the locations around the map so that the entire journey makes more sense than the round trip through the sword coast to come back right where you started. You still have the usual FR problem of "Why are the superpowers of the Realms not stopping the world threatening cult?". Although that is more a RoT problem.
As for the heroic stupid dragon charge, either leave out the dragon completely, it doesn't really add much to the encounter except for a bit of payback you can give it later or you start the raid only after the PCs are inside Greenest. That solution also helps with the first encounter in the adventure where a group of civilians insist that you escort them to safety "right into the center of fighting" instead of fleeing the town by the way you entered without trouble.
Two of my players know nothing about Forgotten Realms and one of my players just rolls with me as DM...in Hoard of the Dragon Queen, I've had "more powerful NPCs" laugh off the threat of the Cult, in preparation for the PCs finding one of those groups having been shredded apart for going in without backup like a typical D&D party. Waterdeep is also getting completely sacked in a brutal, dragon-led assault, which is what's going to bring the various factions and nations together to finally listen to Ontharr Frume and Leosin Erlanther.
I haven't read a Forgotten Realms book in ten years. The only canon that matters to us is what happens at the table.
And the PCs in our group all had reasons to go into Greenest, dragon or no. The Monk's family was part of the Cult until they were betrayed and executed almost to a man by Rezmir, Bog Luck and Frulam Mondath. The Barbarian came from an amazon barbarian tribe that served as smugglers for the Cult, and on her death bed, the Barbarian's mother begged her to redeem the tribe for her deeds. The Dragonborn Ranger had been saved from a raging mob by Leosin, who hid him at Candlekeep. Leosin sent a messenger asking for help, so he responded. Along the way into Greenest, they picked up a Halfling Warlock who just wants to see the entire Cult murdered, no matter what. They went into town regardless. Honor or vengeance wouldn't allow them not to.
Turns out, the Monk wound up leaping from the castle and onto the dragon's back, after they figured out the dragon's heart wasn't in the fight, and he convinced the dragon to abandon the fight. The dragon agreed, abandoning the attack to sleep, not liking Frulam Mondath anyway.
This is now one of the most memorable encounters this group of players has ever had. It also set the theme for this entire campaign early on: They have been slowly weaving alliances together and forging loyalties, while exploiting seams in the supposedly united Cult's allegiances and playing factions against each other.
This has proven to be a great campaign so far, in our experience (we just finished Hoard, and we are starting Rise in two weeks).