Drop bars are fine on gravel
From another 44 year old named Paul with bad knees (caused by excessive running when I was younger).
I have had no problems with drop bars on gravel or dirt roads at all. None. Nada. Put that worry out of your head. The LHT for $900 is a great deal on wheels. You could probably do fine moving the basic componenents of the Trek FX to a good used touring frame from from Ebay - much better than a new hybrid frame. The FX drivetrain is suitablefor touring but my fiancee had a weird rim failure with a bontrager rim on her trek hybrid and I have a healthy skepticism about them. I would get new wheels, which of course moves the used frame + moved parts up to a $650+ project ($400 for good wheels & tires, $150 for frame, $100 for h-bars, shifters, etc).
Frankly, if you have the budget and prefer new to used/project, I would just get the LHT or, given the fact that you don't plan to do hard core touring, the Crosscheck or a Bianchi Volpe. If you like to tinker and personalize your ride, go for a nice old Trek or Miyata touring frame from the '80s or early '90s and transfer the basic drivetrain from the FX. When you can find a frame & fork for under $200, that leaves a lot of budget for quality parts where it counts (wheels, saddle, racks).