Most caulking compounds are junk, and not at all appropriate for repairing concrete cracks. Even if the cracks are routed out and blown clean, the caulking will usually not stay adhered to the base concrete for more than a year or two. And it will always look like dog dung.
The OP's cracks appear to be the result of not having enough control joints (or even expansion joints) in his concrete placement. A wet autumn followed by hard freezing winter weather is often enough to cause them after they never appeared earlier. Corner cracks like the one shown are usually the result of a point load (such as a heavy truck tire) loading an unreinforced corner, especially if a partial void exists under the slab corner. A heavy load will over-stress the concrete, since it has only one direction for distribution instead of an entire surrounding slab.
The best product to use for sealing and repairing such cracks would be a low-modulus, moisture-insensitive epoxy. The term "low-modulus" means it never becomes hard and brittle, but instead retains a certain amount of flexibility. Many such products are available, but usually only at a concrete or masonry supply retailer. Most apron-vest places won't carry them. Doing a search under "low modulus concrete crack filler" will bring up several pages of acceptable products.