Fit the bike for what YOU want, not to satisfy any modern fitting requirement, all of which originate with racing and the need to squeeze the most juice out of the lemon. To get what you want, you need to know where you're starting from, and you need to try different adjustments from that reference point. There's no substitute for experimenting. You're probably not riding with enough racing-type intensity to lift the weight off your arms. Nothing wrong with that, I don't either. For one thing, maintaining that kind of riding position means constant riding and training, and most people can't maintain that kind of commitment over time.
The most common solution is bars higher (up to saddle level -- and farther if necessary as you raise them, like maybe a cm longer reach), AND maybe moving the saddle back a touch (but not necessarily). Sometimes more forward works better. It depends on the geometry of your body more than that of the bike, and you have to experiment.
I sometimes suggest handlebars an inch below the top of the saddle on these forums, but that depends on the context of the question. Most people who want to ride and have fun doing it don't need to get any racier than that, and there's absolutely nothing wrong with having the bars the same level as the saddle if you can get it there. Racers and would be racers are riding full-time on the hoods, which nowadays are where the drops used to be. That's just not practical for most people. But don't get too upright in the process, because then you end up with too much weight on your derriere (not to mention a significant loss of available power). So that might mean a bit longer stem, not shorter as you raise the handlebar. You probably want a back angle of about 50 degrees with your hands on the hoods, and certainly not less than 45. When you want speed or less wind resistance, you've still got your drops at a position where you can actually use them comfortably.
Where do you start from? Well, since you did have a fitting session, you should mark up the bike before you change anything in case you want to go back to it later. Once you've done that, just forget about the whole fitting and fit yourself using the common rules of thumb everybody used to use before the racing fashion. Find a place in your home where you can put the bike on a trainer (or if you don't have one, leaned against a wall by the handlebar while you're on the bike), and where you can put some kind of mirror opposite so you can see yourself on the bike. That helps to see leg angles, back angle, arm angle. Ride the bike after any change, a reasonable distance, both on flats and at least a short climb, and see how it feels. Or go for a longer ride, carrying an allen key tool, and play with the saddle height and setback. It may take a while, but eventually you will hit on the right combination. By that time, you will probably have built up a nice little collection of stems :-)
I would bet that most people end up more or less in the position that would be suitable for long distance touring. Personally, if I have to make a choice, I prefer distance over sheer speed.