I gave up on the cart after the snow took up residence here. Get an Ice fishing sled (Otter sled) they hold a lot of wood and it seems to be much less work pulling a sled full of wood than wheeling a cart.
That's what I use when the snow is too deep for the wheelbarrow. When the woodshed by the OWB is empty and I have to hit the secondary pile I use the plastic tub from a large old ice shanty. The pile is up an incline from the boiler so it slides real nice. Sometimes I have to run to keep ahead of it!