You need to get a good bike shop to fit you professionally to the right bike. Good fit is more important than anything else. You could be riding a bike that costs $8000 but if it don't fit, you won't want to be riding it all day every day for months at a time.
If you're 5'1" then I would suggest first looking at 26" wheeled bikes, since that wheel size will work better with frames in your likely size. It's difficult to make smaller frames in 700C (the larger "road" wheels) without compromising the geometry and creating problems with clearance (e.g. toe overlap and other issues). Besides, 26" wheels are stronger anyway, and such a bike will likely be able to take larger tires - a boon for comfort and ability to take trails that a pure "road" bike might balk at.
Take a look at the Surly Long Haul Trucker - that's a bike in your price range, and it's made from the ground up for carrying a load. Most important thing, though - good fit! People tend to overlook this and focus instead on components or other inconsequential details. Even if you have to drive for a day to get there, it's really worth finding a good bike shop that has the bikes you want to try in stock, and in different sizes so you can compare if you happen to fall between two sizes. Find a bike shop that knows something about touring - a lot of people who work in these places are just road racers and know nothing about touring. But they will still pretend they know what they're talking about! It's human nature. Have a long chat with the owner and ask lots of questions about what they think you should be seeing in a touring bike.
Things to look for include:
o Mount points for front and rear panniers. This includes lower mount points on the front and rear dropouts, and up on the seat stays, and holes halfway up the forks.
o Long chainstays - this will allow you to mount the rear panniers far enough back to avoid heel strike.
o Fender mount points in addition to the ones for the racks - some people don't like fenders, but they are very useful for keeping road muck off you, the drivetrain and the panniers.
o Good gearing for touring - that means three chainrings in the front, usually with something like 26-36-46 teeth, and a good range in the back too. Some people call this "mountain" gearing, since it's often found on mountain bikes. You need low gearing for getting all that weight up the big mountain passes. Even a lot of purpose-made touring bikes have too-high gearing.
o Ability to get the handlebars up around the saddle height - most tourists like the bars a little higher than the racing bike.
If the bike shop guy or gal is talking about this sort of stuff, then that's good. If they seem like they are just trying to sell you whatever they have on the shop floor, then that's not a good sign. If their idea of fitting you is telling you to just take it outside in the parking lot for a spin to "see how it feels" without first measuring your bodily dimensions (leg inseam, arm length etc), then that's not a good sign. Different people with the same height can have different proportions - longer legs and shorter torso and arm reach, or shorter legs and longer torso. These two people of the same height might be comfortable on different size frames, since the length of the top tube determines how "long" the cockpit is. You don't want to be too stretched out, or too hunched up. And it's difficult to tell this stuff sometimes if you're not experienced and just riding a bike around the parking lot. A bike may feel very different after 8 hours in the saddle - any small problems with the fit will become amplified over time.
Get a good bike shop. Find one that stocks the Surly Long Haul Trucker - it sounds perfect for what you're doing, since it comes in 26" wheels and is built for load, the steerer tube comes uncut (allows for getting the bars to the height you want) and the complete bike is around your budget price. See the Surly website for where to find dealers: