Please don't take offense here, but you're not supposed to lean on the bars.
You should support yourself with core muscles, use the bars for balance, bend your elbows, and keep your wrists straight.

I'm using Ahearne Map bars on an upright I built, and find they have a very natural hand position.
Map bars with an 80mm stem place the grips even with the top of the steerer tube (back of stem).
You can adjust your reach forward or back from there with a longer or shorter stem.
A shorter reach stem will make the bike steer a bit faster, but taking your weight off of the bar will make everything more stable and give you more control.
Ahearne+Map Handlebar ? Ahearne Cycles
Another variant on the same bar is the Nitto Jitensha touring bar
Nitto Jitensha B2522 Touring Handlebar 26.0 - Black
here's a good article on "Alt" mountain bars, and you'll see most of them kind of follow this same shape.
Alternative or "Alt" Mountain Bike Handlebar Round Up Not suggesting you buy any of these, but get some ideas for bar shape options.


The next thing to consider are swept-back bar options, Albatross, Northroad or cruiser type bars - these work best with a longer stem.
Take a look at how Rivendell sets up bikes
Bikes

Nice thing about all these bars, they will be able to use your same brakes and controls.