My impressions so far (with limited play time, to be fair) is that Dragons are a bit like a less extreme version of Murlocs. They're good in the early game, when Red Whelp and Waxrider Toggwaggle are crazy good; and have a very strong late game, propelled by Kalecgos, Razorgore, and Nadina (the deathrattle: give your Dragons divine shield minion); but they're a bit more lackluster in the mid-game in between those two periods, due to a lack of buff options (only Menagerie Magician and Zoobot can hit them before the tier 5 Dragon buffer) and the mid-game Dragons really needing buffs to shine (the Divine Shield + Reborn one, the overkill: deal damage one). Also, one of the better mid-game Dragons is the one that gets buffed when your Divine Shields drop, but in the mid-game there's only one Dragon with Divine Shield, and it starts small and, as mentioned, is hard to buff at first; otherwise, only Mechs have Divine Shields (unless you're playing George, anyway). Which pushes it more into the Menagerie comp at that period.
Starting Dragons, transitioning to a part-mech Menagerie-type strategy in the mid-game, and transitioning back to full Dragons if/when you find their late-game power cards might be the way to handle them at this point.