The yellow cloths are perfect for damp dusting - and a hell of a lot cheaper than e-cloths.
The colour tends to come out of them at first when they're put into hot water, but that's not a problem; if it bothers you, stick them through a wash in the machine first.
You soak them in the hottest water you can bear and squeeze them out until they are, as the name suggests, barely damp, then run them along doorframes, doors, skirting boards, the stairs and any other surface. The dust sticks to the cloth, it comes out when you rinse the cloth and doesn't get flicked into the air to breathe in.
Or just buy a bunch of cheap face flannels and use those instead. It's what happened to the majority of terry nappies in the 20 years after the youngest baby was clean and dry at night, after all.
The other things that are good for dealing with allergies are -
New pillows with protectors.
Hard floors.
Bedding washed at 60.
Curtains washed regularly/blinds wiped down daily.
Vacuuming the house every day until it's immaculate and then cutting down to every other/every 3rd but being ready to increase the frequency again if symptoms reappear.
Vacuuming the mattresses daily at first, then putting on protectors once no dust comes off the things.
Doors on bookcases. It's amazing the difference it makes.
A flat microfibre mop - often called a 'Sweep' by contract cleaners - make it damp, push it around daily.
Wiping over any pets regularly with a warm, damp cloth. My cats actually rather enjoy it now, as they seem to think of it as they're being cleaned by their real Mum rather than by their slave
Basically, you need to blitz the place like you've never blitzed it before.
Strong scents can set allergies off, though. I'd suggest you use something unscented for each job at first - there are plenty of brands out there that don't have the pong power of Zoflora (and stay clear of Mr Muscle, whatever you do), leave off on room scents or air fresheners, no aerosols, and if you do use a cleaning chemical, spray it onto a cloth rather than into the air.
I didn't know what the majority of cleaning terms meant until I was in my 30s, simply because I grew up in a fucking hoarder's midden. But now I know, the house is immaculate. And damp dusted.