Was he tested? I ask because you should then know how big of a reaction he had and that may help you determine your course of action.
I am allergic to both cats and dogs, several foods, and most environmental allergens (pollen, mold, etc.) and I have a great allergist!
You should NOT shave or cut their fur because this will only expose the dander more. I actually react much less, if at all, to long haired cats and dogs. I also react much worse to cats (I am "more" allergic)
And also I would hesitate shampooing them more often as this may also cause in increase of dander due to their skin getting drier from repeated shampooing.
If I were you I would get him tested, if you haven't already to get a baseline. The reason I say this is because if he has other allergies this will affect how well he can deal with the dog. For example, if I am around a long-haired cat during the winter when my seasonal allergies aren't acting up, I do not react as quickly nor as strongly to the cat because my body is only dealing with one allergen. If it is prime summer time and I am around a cat/dog my reaction is a lot worse because my body can't handle it all.
A lot of people think that if you have an allergy that you always react the same and react strongly, which is not true. I am allergic to egg whites and some days I can eat a whole egg and nothing happens, sometimes I can eat a cookie that has egg in it and I react horribly and want to die.
If you really, really want to keep your pets I would recommend that you get rid of any carpet/rugs that you have. Vacuuming and especially shampooing can actually make his allergies worse. If you do have carpet and can't get rid of it, then invest in a good vacuum!! And only "dry shampoo" your carpets. Don't let the dogs in his room or near anything he sleeps on, ever! The bedroom should be a pet-free zone. You can even invest in a air cleaner to use in his bedroom at night. Make sure the dogs are brushed outdoors every day and you can even vacuum your dogs afterwards (if they will let you). Make sure he washes his hands after playing with the dogs, or being licked by one. Wash and vacuum your curtains regularly, etc. I would also check into zyrtec or something similar for him or allergy injections, if he is old enough for that.
I'm sorry if this is confusing....but there is so much info and so little time for me to write it. Please email me if you want to talk more about it.
And we have two dogs by the way :)