Assuming you can get the fit issue sorted out (and ideally your LBS will help you sort it out), I'd recommend a road bike based on what you've said.
A shorter stem/longer rise from the fork or a combo of both may help with the back problem. Fore and aft adjustment of the seat could help also.
For all intents and purposes your current bike is a hybrid, and while you can do longer distance on hybrids, I rarely recommend them when people start talking about riding more, riding longer, riding faster, etc.
As for the drop handlebars ... you don't have to ride in the drops, but they're a nice option to have. Riding on the flats of the bar or even the hoods give a somewhat "heads up" riding posture.
And re: the weight limits ... don't put too much stock in them. I've ridden my aluminum road bike when I was over 300 pounds
Welcome to C/A!