A jar of GC10 arrived yesterday. The T3 variant seemed adequate, and the price was a bit less, but it was out of stock, so I got the T4 kind, which has slightly finer metal powder.
So far, I've done 10 boards with it, and none of them have had the shorting problem that I was seeing on previous boards. In fact, my pasting is not really much better, but even with the little slop I make, with gloops of paste sometimes spanning pads, the solder is balling up as it should and I'm not getting short. I have not tested the boards yet, but without the Vcc/Gnd fault, at least it should not be painful to clean up what issues that there might be. Overall, I'm quite pleased so far.
As for my stencil work, I am sure now that the problem is warping of the stencil when I pull it over parts of the board with cutouts. It causes the stencil to lift a bit in spots, causing the deposition of paste in that area to be too deep, and usually sloppy, too. I think my board might be a worst-case scenario for this. Here's a picture of a scrap board inside my stenciling "jig."
The issue is not caused by the cutouts on the inside of the board, but on the big voids on the three "corners" where the board is not in contact with the surrounding scrap holders. I think if I got up some board to tape in those spots, my problem might be mitigated.