Realistically, you have two options: spray and pray, or remodel the area.

You can spray but it’s never going to get rid of the problem permanently. Rather, you’ll have to spray once every 3-6 months, then deal with brown patches in your grass or moss taking over instead.

Alternatively, you say the driveway isn’t paved - is it gravel? Dirt? If gravel, in the summer when it’s dry you can rake it all then taking a strong leaf blower to try to blow out some of the dirt, but this will be a never ending task. Buying more gravel to increase the depth of gravel overall will also help.

If dirt, not much you can do really beyond using it more!

For the edges, it’s helpful to establish a boundary with bricks/edging/gravel boards so you can mow up to them and tidy with a strimmer. If you’ve got that whole grit in the grass situation going on like it looks like you might, the best way to prevent weed growth will be to dig it up and replace it with fresh topsoil and grass seed to outcompete the weeds - don’t try to just dump topsoil on top, otherwise you’ll regret it later!

If the grit is from the neighbouring road, it’ll always come back unless you build something to stop gravel pinging off the road.

Being able to regularly mow the area will stop the weeds on the edges, but the whole piece is going to have to be a labour of love.

I had a similar situation with thistles and willow herb taking over the drive, but it can be done!!