Reporting back: At VCE, the TA took one look at the boxes and said that they would have to go through the oversized baggage scan. She never actually measured them, and didn't mention any fees or overage charges. She put baggage tags on both boxes and sent me over to the oversized baggage scanner to the left of passenger security. They scanned them there and then an airport employee put them on a cart and took them out to the plane.

At IAD, I was expecting them to come out through the oversized baggage claim, but they popped out on the regular carousal.

Meanwhile, the person next to me in the Y line was getting worked over for being a few kg over the 23 kg limit. She was near tears because she had paid for the shrink wrap and was going to have to take it off to get stuff out of her bag. So YMMV.