water in some corners of cement crawlspace
What the contractor is "full of" is excellent advice.

If you really want to spend money like you have a firehose connected to your wallet, dig out around the outside of the house to the footings and put drains there which slope out to daylight (or a sump if you want to be beholden to a sump, and its pump forever) - and coat the walls while you have it open, but the drains are the important part of that job. It's the ultimate drainage retrofit (and SHOULD be done as a matter of course any time a footing/foundation is poured, as that's the only time it's CHEAP to do, since the hole is already there and open), and it STILL goes with fixing the grade around the house when you refill the hole.

If you "dig back on the corners" and coat them, you'll "magically" get leaks elsewhere along the walls - if you are going to dig, dig all the way around and get that over with so you can drain it properly.

It's a far more economically sensible approach to start with removing the source of water trying to drain into the basement in the first place by fixing the grade. There are other relatively less expensive steps you can take as well while doing that, such as laying a plastic or EDPM rubber barrier under the soil that slopes away from the house for some distance, without involving digging all the way down to the footings. But you don't have to be economically sensible, if that's against your principles.

Unless you are 10 feet tall, 7 feet is a basement, by the way. Not one with a lot of headroom, perhaps not one you can finish under some codes, but if you can stand up in it, it's a basement...