There are several breeds that don't produce as much dander or shed as most dogs do.
What you are looking for are the so-called 'hypoallergenic' breeds (cross-breeds). This site has a good list of the breeds that
might be compatible with your allergies:
Dogs Good For Allergy Suffers, Hypo-allergenic dogs
Keep in mind that you will need to contact
reputable breeders (or breed specific rescues) in your area and check out animals individually. Most good breeders will have socialized their animals with other dogs and animals and know which ones do not get along well with other animals.
IF you choose to go to your local animal shelter, contact them ahead of time and explain what you are looking for in an animal. Ask them if there is any way that you could meet possible candidates for new family members in an area away from those who would definitely affect your allergies.
Keep in mind that not only are you needing an animal that is cat-friendly, your cat has to be open to being dog-friendly.
Word of caution, most hypoallergenic breeds that I am aware of tend to be very smart and need owners who are consistent in enforcing the family rules even after the animal seems trained (training never stops). Play is still a part of training. The owner is always the pack leader.
I would suggest taking at least two-three months to find a suitable animal. That way you can explore all your options and take care not to get an animal from a puppy mill or backyard breeder. You want an animal that is as healthy and disorder free (reputable breeders do testing to ensure a greater chance at having a healthy animal) as you can find because an unhealthy animal could be very unhealthy for you.