I did professional fits for awhile in a shop. Your body's balance point relative to the crank centerline will determine how much weight is on the bars. Your upper body should be neutral with no weight on your hands while you are pedaling. The key factor controlling this weight balance point is your saddle setback. In a fit lab, sliding the saddle backwards is almost always necessary. You may even need a different seatpost with more setback to get the proper balanced position.

It is okay to have a very small amount of weight on the bars when you are stationary in the riding position, but it should still be quite easy to get into the riding position while leaning against a wall or with the bike on a trainer, then take your hands off the bars without feeling like you are falling forward at all.

I have a tiny bit of weight on my bars when stationary, but that goes away when I peddle. Like I can completely remove my hands from the bars while pedaling and stay in exactly the same position.

Once you have set up the saddle for a neutral position, only then is it time to find the right stem. Your stem should give you comfortable access the the lowest drop flats, and the main drop position as the primary factor, then the top positions are a secondary factor. Your physiology like your flexibility (and if you have a gut) will determine your lowest position you can sustain. The goal is to make sure all of the muscles in the torso that cross or constrict the lungs are disengaged. You can likely physically peddle at lower positions, but if you can't adequately breathe there is no point in trying to access these positions. If your lower bar positions are too low, you'll never use them.

Your arms should be a 90 degrees in the ideal fit position, if you are reaching, it will cause problems. A little short is okay but too short will make you hunch and compress your lungs.

The first step is always setting your saddle height for leg extension, then saddle setback for a neutral upper body.

The right answer is always a bike fit. However, it could be too expensive.

As for the stems; you could try flipping yours, and see if that helps. If you want to try a shorter one, you could go to a community bike shop and ask for a used one.

Another thing worth looking into is riser drop bars.

The best method is to go out and buy 4 different stems and try them. Honesty.

Bike fits are a scam. Trial and error for your own body and own needs only way to figure this out.

You also need to consider changing your bar width if you alter your stem length/height more than 20cm.

For starters, flip your step up, if it's down. Also, is your bike the correct size frame, and/or is it a racing frame or an endurance frame? Trying to make an aggresive racing frame into a casual endurance bike is not really going to work great.

Weight on your hands is more likely to be a saddle issue. Start by lowering your saddle about half an inch to see if it helps.

One more suggestion is to assess your riding position regardless of what stem you end up going for.

If you're always riding with your elbows locked out and your arms straight, your weight will be on your hands.

If so, think about riding with soft elbows, with your core taking the weight instead. This is overall more comfortable and more aero too so you'll go faster with less effort.