D.B.
We got rid of our allergies and no longer have cat/dog issues. However, we do enjoy the low-shed dogs just from a housekeeping (no vacuum) standpoint.
My neighbors had a wheaton because their son had allergies, and he did fine. They now have Tibetan terriers. Other neighbors got a Havanese for the same reason. We had a Cairn terrier before we got rid of our allergies - I can't say she was entirely non-shedding but she wasn't like the golden retrievers or most other dogs where all you do is vacuum. A friend with asthma came over to visit to see if she could handle a Cairn, and she was fine while she was here but not perfect the next day - not sure if it was due to the dog or not. We now have a terrier mix from a shelter - not sure of her exact parentage but she appears to be Cairn/Affenpinscher/Dandie Dinmont terrier - and she doesn't shed, even less than our purebred Cairn.
My niece got a Portuguese water dog - same reason. Other friends have a golden-doodle and another has a "schnoodle" (schnauzer/poodle). Lots of dogs mixed with poodle are popular for allergy sufferers.
I think you could consider spending some time with a dog before you make a decision, or you could consider nutritional allergy abatement (no shots, no meds) like we did, or both.