I think you can just put cardboard down on the paths now and cover them with straw or other mulch. It sounds daunting, but I did it for my garden, which is almost the same size, and it wasn't too bad. The little bits of edge grass that remain will be easy to pull up once the rest of the sod is dead. An alternative is to edge every bed with plastic or metal edging, but I wouldn't trust it near soil that's growing food. You also still have the problem of grass clippings getting onto the leaves of plants when you mow.

Once the grass is dead you can put down some kind of more permanent path. I just add new straw on top of the old every year to keep it thick enough to suppress weeds. Every couple of years I put new cardboard underneath if it needs it.