The solution is a second set of footwear that is lightweight and inexpensive. Perhaps sandals, or canvas deck shoes. When you arrive at the campsite, change shoes and set your hiking shoes out in the sun to dry and to air out. Wear the lightweight shoes as you set up camp, cook your evening meal, and so on. If it's raining, hang them up under your tarp (rain protection) - believe it or not things do dry a little this way.
It should not be moist in the tent. Typically it is a little warmer in the tent than out, and you should have ventilation. Let both sets of shoes spend the night in the tent with you. Your hiking shoes should not get worse over night. In the morning, put on the lightweight ones and first thing you do, find some sun to get the hiking shoes some more drying time while you prepare breakfast, pack up the camp, and such. One of the last things you do as you leave is switch from your lightweight camp shoes to your hiking shoes for the day, putting the lightweight shoes in your bag.
Continue each day and your shoes should stay dry and well aired out. The alternative shoes don't need to be hiking shoes, so they don't need to be expensive or heavy at all. Also, your feet will be healthier switching into dry shoes (and even better, dry socks too) twice a day.