It'd take a LOT of soil to build that back yard up to get it to drain to another yard.....I have used "dry wells/pits" in the past....I would use more than one.....find the center of each low spot and sink a hole at least 12" wide by 4' deep.....you can use the cardboard cement tube forms to wrap a landscape fabric around.........put it down in the hole, fill with a 57 stone and then pull the form out a little at a time as you fill the hole up.....or.....I have used a small back-hoe to dig a 4'x6' pit, lined it with a landscape fabric using sod staples to pin it to the sides and bottom, and fill it back in leaving the top few inches covered with soil.....
of course, if you have a very wet or heavy rain where the water table rises in the ground, the success would be limited....
Also, since you're already taking these steps, consider installing a pit with a sump or septic pump and pump excess water to where you want it. Yes you'll need to run electricity (underground) and then install an underground pipe to where ever you want the water to go but, that beats the heck out of wading through ankle deep water when it rains.
I had a shop built for my hobbies and toys, the contractor took the soil down to well below grade level, so when it rained, the shop was often threatened with flooding and actually got wet once. Once was enough for me. I rented the equipment and installed a French Drain and then dug a trough out to the ditch that runs along the road I live on. I installed drain pipe and covered it with dirt. Now, I have no water issues. Of course, I was able to use gravity to my favor for drainage but, once you get that water over the "high" point with a pump, the same principle would work for you. This assumes you have somewhere to pump that water to.
