MichaelW just about took the reply out of my mouth.
Cannondale has their T series bikes which are very well liked for touring and would work superbly for your other riding. You can find anything from the T800 to T2000 in your price range used and in fantastic condition. Same goes for Trek's 520. These bikes are always on eBay. Here's an example:
T800 auction
The Fuji Touring and Jamis Aurora are easy to find new for less than $1,000. I've seen them both used for coast to coast loaded touring.
You can build up a LHT for 1,000 and I had a friend order one from a local shop fully built up for about 1,100. The other bikes listed above are going to be more responsive for unloaded riding though.
If it were me I'd lean toward a Cannondale. They can really take a beating. Here's an example:
Mark Boyd's Cannondale and tours
Good luck and have fun finding your bike.
Ron
Wasatch Mountain Range, Utah
Vermont, New Hampshire and Maine x2 in 3 days.
Our Touring Journals
The excellent website we use for our journals is having a rare problem. Hopefully the owner gets it fixed inside of three days.