Mouldy Seedlings?

The toilet rolls have developed white fluffy mould on the outside. It has been raining for weeks here, so the air is moist. Hope it doesn't spread mould around the house or cause problems in the garden when I plant them.
The broccoli seedlings have 2-3 tiny leaves and a hair-like root just showing at the base, and the beans haven't sprouted at all yet, but I started to dig one up and it is germinating, maybe I planted them upside down? Next time I will soak them before planting.