Not using a fan is exactly why you are feeling so awful. Seriously, you need one. There are loads of articles about how overheating can do you damage, and with no air movement at all indoors, overheating can happen a lot more quickly. I use a fan all year round. It gets turned on further into the ride in the winter but it is still needed every ride.
Zwift won't disqualify you for being out of category but Zwift Power will which is like the official tool used for races. Just create an account and it will work out what cat you meant to be in. I was high C until I got my kickr, and now I am low c.