Check out some of the threads in the foaling section for blow-by- blow photos of mare's udders. Especially Sunshine's. They got her with no idea of due date, also, and her vet said "a couple of weeks" which turned into 6 months!! So, don't believe everything the vet says.
Your mare should have Rhino vaccinations at 5, 7, and 9 months, and about 6 weeks before foaling, get a full vaccination set, so that she passes that immonity to the foal.
It is probably too late to do an ultrasound, but the vet can palpate a pregnancy. It willl only be a guess as to due date, though, so I would start vaccinating her ASAP.