I have in mind something I can put up on my shoulders or around my waist. Not necessarily to strap it on; something I can doff at will, but that will stay secure, like shoulder hooks or something. In transit, I want my hands free to be able to other things, or carry other things, whatever. But I don't want to be unable to drop the cart and run or dodge if I have to without undo manipulation.
A cart makes a lot of sense, especially in an urban environment that is for the most part still intact. Even in rugged terrain it could be helpful, but within limits. Maybe there'd be a way to rig up something you could pack on your back if you had to, and when the going is a bit easier you can put it down on integral wheels.
It sound's like your looking for a rickshaw, towed hand truck, or a wheeled travois type setup where the cart is attached to you via a harness. EDIT: removing all the DIY harness stuff and sugesting you get a commercial sled "pulk" harness.
There's a variety of
marine quick release snap shackles that can be undone quickly and without dexterity that you could use to attach the cart to the harness. They tend to be stiff and typically have a cord loop attached to them for ease of use.
EDIT: Be careful with the 99 or 100 series masks. The effort of pulling air through the filter adds additional load onto your heart and lungs. In addition to N series, there's also R and P ratings. I used P-95 rated 3M Tekk Protection Paint Odor Valved Respirators (Home Depot) when I had to paint my house, and they made a big difference (I'm sensative to a lot of organic fumes)