A Touring bike is meant to carry people and loads over a distance. It tends to be more comfortable and have a more relaxed geometry and riding position. A cyclocross bike will be a little bit more aggressive in its handling and rider position. Usually it will be faster than a touring bike, but not as aggressive as a racing bike. It is considered somewhat a jack of all trades.

I think the drop handlebars will give you more hand positions than some of the flat barred bikes mentioned which I think is beneficial over distances. However if you are riding 20 miles in the city and stopping at lights, intersections, and so forth the handlebars may not matter. If you will have a route with 3-5 miles here and there of straight riding than drop handlebars would probably be beneficial.

Starting at $800 you might find the Surly CrossCheck and Long Haul Trucker just a bit over your budget. You might find a Windsor or Fuji tourer right around that price. I believe you can also look into the Soma Double Cross or REI Novarro (I believe those are two others). Do a search for Touring Bikes or Cyclocross Bikes to give an idea. But as always ride a variety of bikes.