Likely it would cost enough that it would be worthwhile to rethink your budget and just get a touring bike. Getting new wheels is about the priciest upgrade.

That said, my wife rides a Giant Escape. Slightly different than that model. We've only gone on one, multi-day trip with it, about 40 miles/day and staying in hotels, so no camping gear. She likes it, but has complained that 40 miles on flat bars with no alternate hand positions can be a pain, literally.

It wouldn't be my choice, but I've toured on bikes less suited to the task. My preference with that budget would be to look used. Look steel (I know, I know. Aluminum's fine. I still prefer steel). And don't pay too much if you're going to want to upgrade the wheels because you need to set aside money for that.