You could also make a difference between being in the line of fire, and being at the point of impact of the breath weapon. The latter could deal bonus damage.
Flames could also spread outward each round from the original tiles that were hit. At a speed of 1 tile (5 ft.) per round, up to a maximum determined by you. But if they are in an open field, the fire would probably keep spreading. I would focus less on making the damage of being on fire really high (just keep it at the normal 1d10 per round) and more on just engulfing an area, leaving very little safe spots where they aren't set on fire. I would always try to keep as much of the original rules intact, if only for the sake of consistency.
Then you just say "Drogon's using Quicken Metamagic" to cast and breathe at the same time, and you're done
Even that is over complicating things. Just state that a dragon can move its breathweapon while it is spitting fire (up to a 180 arc around itself), and any of the tiles hit are also set on fire.
Consequently, I'm sure you can see why I'd have a very strong aversion to rape in my entertainment.
I don't think anyone needs to give a reason to why they have a strong aversion to rape. I would assume thats a given. But the show doesn't actually feature it all that much come to think of it. I remember only 3 outright cases, and one attempt across all the seasons. Given that the show is portraying medieval times, thats probably pretty moderate. The show actually has far less of it than the books (because the books outright say that the Ironborn engage in it on a regular basis, its one of their favourite hobbies). But there were some scenes that were shown in the show, that were more consensual in the books. Jamie and Cersei in the Septon of Balor for example, or Dany and Drogo in season 1. But the show also features a lot of torture, which I find even more unpleasant to watch, truth be told.
Of course we all choose what shows we prefer want to watch. There's plenty of crime shows that I never watch, because I hate drug culture and organized crime stuff, even though some of them are probably great shows, but I just don't like the theme.
I'm sorry you had the D&D experience that you did. One of the core things my group has always done at the start of every new campaign, is to establish some clear rules regarding what is and what is not off limits. Will there be mature content? If so, do we just fade to black, or describe what happens? Does anyone have objections to certain themes? I remember the Book of Vile Darkness for 3.5 warned its readers that a lot of the stuff in the books was probably not for everyone. Its one of the few times I've seen D&D get really dark. So this is perhaps a good warning for all players out there (old and new), be sure to communicate what themes you do and do not want to see in the campaign you are playing, before you start playing. If the campaign is more along the lines of Game of Thrones, is everyone on board with that, or are there objections? Be sure that no one feels uncomfortable with certain themes in the campaign. Specifically when it comes to themes of rape, torture, but also religion.