A couple years back I used an edger to carve a nice, clean lawn edge
along my driveway and along the street. Over time, the crevice has
grown larger and now accumulates sticks, stones, weeds, etc. Long
story short, I want to fill it in and have grass grow there again.

Here's my plan. Tell me if it sounds reasonable. First, I'm going to
take a hoe through it and clear it of all debris. Then I plan to fill
it with a mixture of turf soil and grass seed and tamp it down to
roughly the level of the existing turf.

Will that work or will the soil wash away with the first heavy rain?
Also, is that something I can do this time of year (I live in Central
Ohio) or should I wait until September?

Thanks...