The style of bars you referring to are drop bars.  

Hybrid / city / multi-use bikes now come with both styles of handle bars.   
Not endorsing this bike.  It is just an example. [MultUseWithDropBars][2]  

One thing to consider is a used cyclocross and put touring tires on it.  It has drop bars and will take larger tires.  You can get great deals on used as many racers upgrade every season or two.  And this is the end of cyclocross season.  Or a used touring bike but you don't get as good of deals on used touring bike as you don't have racers flooding used market.  But if you don't know bikes then you can get burnt used. This is a cyclocross set up with 35mm touring tires that I use for training and crashing around the city.

![enter image description here][1]


  [1]: https://i.sstatic.net/U3LUx.jpg
  [2]: http://www.specialized.com/us/en/bikes/fitness/awol/awol