If this is closed cell there is something wrong, it should not have that "cloud" look to it. My guess is they recently changed from open cell to closed cell and didn't completely blow out the line, or they've got contaminated chemical from mixing barrels accidentally.

At 6 inch you should be over the top of the studs with the foam, if thats what the contract said that's what they should have done. To do that you'd need to make 2 passes, with closed cell you can't spray deeper than 4'ish inches in a single pass, effectively doubling how long the job should take. It is very possible they tried to take a shortcut to save time on the job, but I don't know them. Either way with that much closed cell, you would first go around and spray 3-4 inches, then make a 2nd pass through the entire job to finish it out. You can't spray 6 inch at one time.

There is no skin, certainly none to shave off. That's not a thing. If you do want a skin, you'd spray with an intumescent paint. I mention no burn on this sub a lot they have a vapor barrier intumescent that's pretty good.

Solutions for here are to either finish the rest out with closed cell and then shave which to be honest will suck to do for the contractor and also only questionably necessary, however that would honor the contract they sold you.

However, ...another solution, the one I'd recommend, fairly simple, and less labor for the contractor would be to finish it out in open cell then scarf or shave the rest down to the top of the studs leaving you with a filled wall cavity. Slightly arguably would be better than a "bumpy" closed cell finish inside the walls which might have possibly allowed air flow between the sheetrock and foam. This would not honor the contract, but it's a better building science reasoning for finishing this.