Not sure about the allergies - depending on dog, but at the most severe it would be contact (hives), runny eyes, itching and wheezing. Very attractive . That was with my Mum's Collie/ alsatian X. With others it's just a slight itch/ slightly sore eyes. I think if it's mild, then I should get used to it. Will have to go and rub my face on any prospective pets!
There are a massive range of dogs who don't shed their fur and are suitable for allergies. The 'designer' breeds are often non-shedding, but this is not always the case. Most breeds have their own rescues run by volunteers who do an amazing job. I am currently salivating over a couple of poodles on the Poodle rescue website (I know I am only allowed one dog, but....). The kennel club lists these rescues by breed.
Second Lizcat's recommendation to contact breed rescue if you have a particular breed in mind. Our practice was involved in rehoming an Airedale last year, when the elderly owner very sadly became too ill to keep her, and Airedale breed rescue were wonderful - put us in touch with a selection of pre-vetted Airedale-loving homes, and she ended up going 300 miles to live with an old lady much like the one she'd left (but healthier!), with the backup of the old lady's Airedale-loving son in case of trouble. It was a match made in heaven that breed rescue made possible - they can do a wonderful job. If you go that route, though, you may not get a dog instantly, as they will rightly prioritise the right home for the dog over finding any dog for the homes that want them, if you see what I mean. Hope it works out for you.