Honestly the bike you have looks pretty good. If it is reasonably comfortable and works well, I would try installing some narrower semi-slick tires first as suggested in another post.

While I love my entry level Giant Sedona comfort bike, Low end, which might be what $800 will get even from more popular brands may not be a lot better then what you have now. Mostly different. Which can be a good thing.

Hybrid bikes like Giant's Roam or Escape might be a good bikes to check out if your unsatisfied with your bike after trying it out with quicker tires. These adventure bikes work well on paved roads, gravel, and many trails. I believe they have low enough gears for pretty steep hills and should be within your budget. Trek and Specialized are other brands I trust that probably have similar bikes to Giant's Roam and Escape.