I'm glad you've 'bounced back' a bit today. I'm sorry to hear you almost sent yourself to your local ER. That stuff is nasty!
Anyways, get better first. Yes, it's VERY likely that your tiles have asbestos in them, or at worst, it's in the glue. However, from what you're saying, there doesn't seem to be any hazard. The hazard lies in the asbestos becomming airborne. This usually occurs when one takes a power saw or somethin' to the floor and creates a lot of dust. This 'dust/residue' should be OK. It's not like you managed to 'separate' the asbestos from the flooring, or something.
I say to start by taking a damp sponge mop and mop up the floor as you normally would. Don't use a lot of water - Just a damp sponge. Keep wringing it out and change the water if it starts to look cloudy. You can dump it curbside or in a remote section of your lot if you feel you have a hazard (which I don't). I wouldn't go watering my plants with the 'spent water' though.
You may find that your efforts are fruitless. This may mean tha the 'finished surface' (for lack of better words), is ruined. If this is the case, I don't know of anything that can be done to 'restore' your floor. Beings it's from the 1950's, I'd consider getting a new one in time. Maybe others have a 'way' to restore a floor such as yours.