My mare has this too–it started about a week and a half ago (she is also 281 days along), and I called my regular vet, since I was concerned.
There was no “udder”, per se, but on one side, there was a long, tubelike swelling in front of and behind one teat. My vet is NOT a repro vet ;), so I called down to ERC and talked to Robin, hoping that the vet down there might have some thoughts. When I described it, Robin said “that’s her milk vein”, explaining that the milk vein that feeds the udder often becomes swollen and appears tubelike at around that gestational stage.
WHEW! Since it was only on one side, I wasn’t concerned, there is no obvious udderly appearence, and her teats are small and dry. It is bigger than Robin said it usually is (the diameter of a finger), but today when I checked on her, there seems to be more edema on the other side as well. 
It seems to be mostly in front of and behind her teats, and is insensitive, though is a hard swelling.
My mare IS a sweller! She loves to puff up with edema, especially when it’s warm and muggy, so I am HOPING that this isn’t something I should be worried about, but plan to call a local repro vet on Monday and see whether she can either reassure me that this is okay, or come out and take a look at her.
(I love my own vet, but she is NOT a repro vet, so I would prefer not to take any chances…)
If anyone else could weigh in with their experiences, that would be very helpful! OP, how does your mare’s udder look now, any changes?