As paddleboy notes, the part that makes sense (5'8" woman riding a 54 cm bike) is that your legs are proportionately longer than your torso. If you need to lose some weight (like so many of us), it makes sense that you need the bars up near the level of your saddle, because your tummy gets in the way of your thighs when you're pedaling.
I'm grasping at straws still, but I'll go further out on a limb anyway. When you bought your Madone, were the bars slammed down against the headset, and there were no spacers above the stem? If that's the case, I'm guessing you bought the bike off the showroom floor, and the person who set it up for display cut the steeerer so there was no vertical adjustment possible (because it looked more like a racing bike that way!). It's possible the biggest part of your fit issues has been trying to get the bars up with stem swaps, perhaps even changing out bars to a shorter reach model to pull the ramps back toward your saddle.
If I haven't sawn off my limb with me on it yet, bottom line is you need to catch the bike coming out of the box from the factory before the steerer is cut. That will let you mount the stem and bars higher and closer to the saddle, for a reduced reach. Most bikes come from the factory like that -- just don't let anybody get near it with a hacksaw or tubing cutter before you get your hands on it! Most normal touring bikes should be amenable to fitting if this is your case. I suspect it'd be simple to swap in a shorter stem with a similar angle, if that's necessary ($30 vice $75), and you shouldn't need to change out the bars ($$ in labor to re-install shifters and re-wrap the bars).
Note, too, that many tourists prefer to sit up a bit compared to a normal racing fit, both for better views and to support longer days in the saddle. If you ride with your hands on the brake levers, or even on top, you don't need to go to as much trouble raising the bars as if you plan to spends your trip in the drops.
