Lower bars will have you leaning forward, yes. As you lean forward, your upper body CG moves forward. If your saddle fore-and-aft remains the same, obviously this will put more weight on your hands. There is a test for balance: you briefly take your hands off the bars If this causes you to slide forward on the saddle, you need to move the saddle back. You also may need a longer stem. The downforce of your hands on the bars prevents your torso from rotating down. Thus your arms are acting as a lever. The longer the lever, the less force is required to generate the moment which supports your torso. Hence there's a sort of rule that your upper arms should make a right angle with your torso. I.e. you may need a longer stem. Observe in a mirror or have someone take a photo.

My bike fit instructions are here: How can I fitting my bike (Thread name was not my choice.) When one changes one thing, other things also have to change. Thus the whole fitting thing is a process of many changes, and it takes time on the bike to really tell if a particular fit will work for you. My standard is one 3 hour ride, minimum.