I believe this is the "Magic Time" series by Marc Zicree and a different collaborator for each book (Barbara Hambly for the first book, which is why I picked it up). It has been many years since I read it, but IIRC, a lot of modern technology stopped working, so it was definitely a disastrous/apocalyptic setting, and there was someone who turned into a dragon.
For rising young lawyer Cal Griffin, it's just another day in the Big City--until the lights go off...for good. Suddenly packs of pale crouched figures are stalking the darkened subways, monsters prowl Times Square, and the people all around Cal are...changing. Similar weirdness is happening everywhere, from the dank, cold heart of a West Virginia coal mine to a remote lab in South Dakota--where a team of government scientists has unwittingly invited something catastrophic into the world--to the highest levels of power in Washington, D.C. And Cal Griffin is not the only one struggling to comprehend the surreal, devouring chaos surrounding him--nor the only one who will be forced to accept a new role in this brave new world of nightmare and wonder. For the forces bled from the stilled machines are fueling a consciousness both newly born and ancient--and more than one unlikely hero will be needed for the titanic battle between the darkness and the light.
This first collaboration between talented writers Marc Scott Zicree and Barbara Hambly is the beginning of an innovative fantasy series. An enthralling and entertaining read, Magic Time mixes supernatural forces and elements of classic epic-quest fantasy with modern technology and contemporary themes in a fascinating "What if?" scenario that presents a world where science is turned on its head as magical forces -- and strange new creatures -- blossom in its place.
Goodreads, Magic Time