I think there are three options to look at that meet all your requirements:
- A touring bike
- A Cyclocross bike
- Build what you want starting with an old steel frame. (70s or 80s road, or hard-tailed mountain bike)
Note when shopping that a good rack is going to run you $50-$200 (and you want a front and back, so a total of two racks) and good panniers seem to run $100-$200 for a pair (and you may want two pairs). If a bike comes with good racks and costs more it may actually work out cheaper for you.
In general, you may want to look for bikes that have 26" wheels instead of 700c wheels. Since you're 5'4", fitting the larger wheels on a bike involves some frame geometry compromises. Also, if you're planning to do touring in other countries, apparently the 26" wheel size is easier to find in Asia and south america.
Also consider having the shop replace the stem and handlebars to get the fit right for you. You may find that the bikes come stock with handlebars that are a little too wide for you and a little too far forward to reach comfortably. (however, fit is tricky since there's a lot of details and height isn't enough)
There's a lot of sub-variants in the general touring bike category, so not all meet these descriptions
- Touring bikes are specifically made for carrying heavy loads using a front and back rack. Holes to bolt the racks, heavy duty frame, heavy duty wheels, etc.
- Touring bikes are generally heavy-duty enough to handle a bit of off-road paths and many people include fire roads and other dirt or gravel roads in their touring.
- Touring bikes usually are designed to handle fairly fat tires (for comfort over long distances and rough roads) and could have knobby tires mounted easily enough. Or fenders, though fenders limit how fat you can go on the tires a little.
- Drop handlebars are common on touring bikes, though many people opt for various other bar styles
- Touring bikes work great as a daily commuter (I use one that way)
- Finding a touring bike for somebody who is 5'4" could be a little tricky
- Many touring bikes are above your price-point by a few hundred US dollars.
As far as specific models, a few people have mentioned Novara bikes, which are sold at REI. I ride a Surly Long Haul Trucker which I believe does have a frame size option that might fit you and I think still retails for about $1100 (and since it's distributed by QBP, most bike shops can order it).
Cyclocross is a sport that involves a mix of off-road and on-road. Usually reasonably smooth dirt and/or mud. No jumps. The most expensive cyclocross bikes are optimized entirely for racing and have no rack bolt-holes, but the lower-end cyclocross bikes often have bolt-holes for fenders and/or racks.
- Cyclocross bikes are designed for light off-road. Because of that, they're likely to be tough enough for what you want.
- Cyclocross bikes usually come with tires that are slightly knobby, and often have clearance for slightly larger tires than that (so that muddy tires can still roll)
- Drop bars are the standard on cyclocross bikes
- Once you've added a rack (or two, or basket, or rack and basket) and fenders, these make great commuter bikes. I've definitely seen people using them that way. For commuting or touring I would replace the stock knobby tires with a smooth puncture-resistant tire with reflective sidewalls (Schwalbe Marathon something, Specialized Armadillo, etc). For more dedicated off-road usage you might want to replace the tires with something more knobby than the stock tires.
Specific models: The Surly Cross-Check should work, it's very similar to the Surly Long Haul Trucker I mentioned above. I've seen Specialized Tri-Cross bikes for under $1000.
Right now is the very beginning of Cyclocross race season in much of the US, so if you're in the US it's likely stores will have some on the floor to check out.