No need to till. Just add soil, making sure that the finished grade drains. New soil doesn't mean any grass will grow if there is too much shade. You can only grow plants that tolerate the conditions you're offering.

If you plant grass for shade, keep in mind that it has to be reseeded regularly, much more so than grass for sun.

A couple of truck loads doesn't go far. Get a rough idea of how many square feet the area is, multiply by .33 (for 4 inches of soil), and divide by 27. That's how many cubic yards of soil you need. A quarter of an acre 4 inches deep is a lot of topsoil.

If it is 'very shady', I assume that shade comes from trees growing in that area? If so you need to rethink bringing in truckloads of soil to raise the level. Increasing the soil level....even a couple of inches.....will effectively smother the tree roots and kill them. It may not happen immediately but it will happen!!

Shade, drought, and dog tolerant. You're the first one this year to ask for the moon, but you won't be the last.

It would be much easier to drain the swamp than fill the swamp. Cut a swale into the low side of the yard to let water drain out. Or hire a landscaper to bring in a tractor and box blade to reprofile the entire area. For one acre it should take less than a morning with the right tools. If the landscaper pulls up with a skid steer or bobcat, send him away before he unloads the equipment. It will take weeks to fix with the wrong stuff, and remember, you're paying by the hour.

If the soil is hard or otherwise horrible, that will essentially fix itself or can be fixed easily once stuff starts growing. If you cover the area with mulch, like Mother Nature tries to do every autumn, the soil will soften by itself. Otherwise just taking proper care of the soil will help soften it. Deep and infrequent watering is the proper care I'm talking about. Then to make it still better, drop some organic fertilizer several times a year at a rate of 15 to 20 pounds per 1,000 square feet. I usually use alfalfa pellets or corn meal depending on local price. In your area corn or soybean meal might be easier to find at good prices. Corn should be well under $10 per bag at the feed store. For a full acre you might consider a bulk delivery.

To figure out how square footage you need, use this tool:

https://www.mapdevelopers.com/distance_finder.php

It allows you to create points over a google map and can tell you the distance. To figure out how much soil you need, then use this tool:

https://www.soilcalculator.com

It allows you to enter dimensions and calculate in cubic yards.