This might seem like a stupid question, but do you need to equip a shirt and shoes on your char in order to avoid getting sick, or is it enough to just keep a pair of shoes, shirt, and/or winter coat in your inventory throughout the game?

I always equip is, just in case. It's very easy to catch a disease if you don't follow certain precautions, like wearing winter clothes in winter or camping without a bedroll.

Why would you not put the shoes into the shoe slot? You cannot put anything else there anyway.
Shoes or shirts in your inventory will not do anything. You should not wear shirts though. The only thing that belongs into the torso slot is the best armor the class can wear.
In RoA 2 I always wear the green robes in the robe slot and cary a sleeping bag in the inventory. Winter clothes would be better in the winter and bedrolls in the summer. But I do not like to change my equipment every time the seasons change.