I can assure you they clean and sanitize between guests, and they do sanitize while you are still the guest. Linens are bagged separately, as are towels. If there are large, unknown stains of any kind, they are specially bagged. If a guest notifies that they’ve been sick, the room is cleaned and sanitized and all linens, towels are sent for special cleaning or discarded. Many of the housekeepers glove or even double glove.
My suggestion is to contact the front desk of the resort you are staying at and tell them of your issues. I have a friend who is allergic to just about anything – seriously allergic – through multiple stays at all different resorts, even DVCs, there is an annotation of allergies. She calls the week before to talk to the front desk, who enters it into the computer for housekeeping to see. If there is a throw and throw pillows in the room, housekeeping removes it. Housekeeping puts cleaned sheets over the furniture. She prefers to sleep between 2 sheets and they remove the blanket and comforter – she does bring her own blanket. She also brings her own pillow.
Stay in touch with your resort and YOU tell them all that you need… most resorts will accommodate that. If they say no, call higher for a manager. It can be done at Disney. My friend has been coming to Disney for a very long time and Disney has done everything they can to help her. It may be tough at first, but stick to your guns (an expression) and you’ll receive nevessary acommodations. Disney will try for you.