This photos was from a month ago. The grass isn't as lush and is looking quite sad and sparse at the moment.

I would not bring in a single yard of topsoil. 10 to 12 will certainly ruin your drainage possibly causing water to flow up against your house leading to rot and possibly termites and ants. Instead bring in a tractor with a box blade to level the surface and maintain your current drainage. You might have to tear down a portion of fence, temporarily, to make room for the tractor to get into the yard. Call some landscapers to get estimates. If any of them suggest using a skid steer or a bobcat to do the job, move on to the next landscaper. That is a one-morning job for a tractor and a 3-day job for a bobcat.

With mention of bigleaf maples and laurels, I assume you are in the PNW?

Yes, you can bring in soil to level and I would advise that grading or leveling using any power equipment is not necessary. And you won't need anywhere close to 10-12 yards. 3 yards will provide an inch of soil over the lawn and that is plenty. Fill any major divots first, firming the soil well but avoiding compacting it. Often just a strong spray of water will do all the firming necessary :-) Then spread/rake the rest out evenly. It may not provide you with a perfectly level surface right off the bat but is something that can be done annually or as often as needed to achieve the results you want.

fwiw, it looks like most of the lumpiness or bumpiness is caused by very sporadic and patchy lawn growth. If you get a healthy lawn growing, I think a lot of your unevenness issues will just disappear.

GG, what is the point of adding topsoil? And why would you bring in an inch every year? It would not be long before the soil is up to the windows. Here is a picture of a yard where they used 1/4-inch of topsoil every year...for 40 years.

The stop sign area is mounded about 15 inches up the sign. They even had to put in metal borders to keep the soil from washing out onto the sidewalk.

When I was in college I worked for a finish grading contractor in SoCal. We fixed a lot more problems by hauling soil away than we did by addition.

I agree with garden gal.....just add enough top soil to level the lumps and bumps...and to give the grass seed a place to settle into.
And by "do it every year" I am sure she didn't mean to bring in 2 or 3 years of soil but just enough to fill any low spots and add a little seed.
You also might even get away with not adding top soil just rent an aerator...the kind that pulls out little plugs of soil......then add seed and water very well. In a couple of weeks all those little plugs will melt into the existing grass, leveling it and providing a tooth for the seed.

@gardengal48 (PNW Z8/9) Yes, just 20 min outside of Seattle. Because I'm a total newbie, forgive me if this is a silly question.
Would you not suggest placing soil on top of the existing grass then, and basicalling flat/level up to it and seed the bald spots? I honestly thought we'd mow very short, level with topsoil and broadcast seed throughout the entire yard. We also need to align the soil to the foundation. Some spots show more foundation than others and it would be nice to have it be a consistent line.

Thanks everyone for the input. I'm taking it all in.

Yes, that's essentially what I was saying. You just don't need anywhere near the quantity you were first quoted. You don't really want to raise the soil/lawn level.....just to even it out.

Do the lawn first. You are rapidly coming to the end of the window for spring seeding here (May 15th), so focus on the lawn first and address the soil level in any planting beds later.

20 min outside of Seattle

Turning this over to you, GG. This is your wheelhouse.

There should be many posts here on Houzz regarding lawn leveling going back 13 years. Most apply to bermuda lawns, but the basics are there for modification to other turf types.