Well we got it cut. I'm pretty sure Brian got down through the slab. He felt it would be best to follow the existing cuts, so the result isn't pretty, but I'm hopeful it will do the job.
I spent part of thursday and part of today trying to clear the slurry out of the cut. I used a water hose + shop vac, trying to force the stuff out of the cut. This proved to be surprisingly difficult. I'd thought I had 90% of it on Thursday, but I looked at it again yesterday, and it seemed I'd only removed about 50% of it. I think I got most of the rest of it today. Probably about as much as I'm going to be able to get - I used water pressure, a shop vac, and an air compressor to try to force the remaining gunk out of the cuts.