You're running into a classic problem with slope seeding, and water flow is definitely the culprit. I dealt with this myself on a hillside patch - the seed was washing downslope faster than it could germinate. What helped was laying down erosion control blanket or even straw mulch over the seeded area; it slows the water runoff enough to let the seed settle in, and it keeps the soil from drying out too fast between rains. If the driveway runoff is a bigger issue than just slope, you might need to redirect that water or create a small berm to channel it away from your seeding zone. Another approach that worked for me was watering lightly and frequently with a sprinkler rather than relying on heavy rain to do the work - gives the seed time to take hold before nature dumping an inch on it all at once.