I have slight asthma and terrible allergies, which pet danders seem to set off. Despite being allergic to nearly everything under the sun, I have no problem with keeping birds. However, I don't own any of the dustier species like cockatoos and cockatiels, so I'm not sure how I would react with them in the house. Though, I have worked extensively with all species (including 'tiels and 'toos) in the past and haven't noticed any problems.
However, each person is different!
If you are allergic to the point that it will cause trouble, you will likely know very quickly - and probably would already know by now. So I'd say it's probably perfectly fine for you to interact with the birds without worry, though I would keep an inhaler and antihistamines on hand just in case.
If you do have reactions, there are things you can do to prevent them. A doctor (ideally an ear, nose, and throat doctor) would probably be able to best walk you through your options, but antihistamine pills, nasal sprays, and eye drops are all things I've used liberally in the past to combat my allergic reactions, and they're all readily available over the counter.
As for the birds, you can invest in a good air filter and keep the humidity a little higher (~50%) to try to keep the dander out of the air. Frequent spray baths and regular cage cleanings (and misting the bottom of the cages with water before changing the papers) will all help a bit with keeping the dust down.
In general, people consider cockatoos, cockatiels, and african greys to be the dustier species.