smudge
Up the Albion!
A friend of mine's son-in-law ruined his lawn by applying "Weed & Feed" without reading the instructions. He ran out of the stuff, went and bought some more and put that on too. His lawn looked similar to how you describe yours. It was full of bare patches where he had over-dosed it, which is exactly what happens when dog bitches urinate on grass. Basically, fertilizer is piss!
We forked over the dead patches to the depth of an inch or two, spread grass seed over it and diggled it in with the side of a trowel. Grass seed needs to be worked in to just under the surface of the soil.
Three weeks later, you couldn't tell the lawn was a mess previously.
It may get a little colder in the next few weeks, but sowing grass seed now isn't a bad idea. There is a saying regarding grass seed - "Better in the ground than in the bag."
As Lammie said earlier, just lightly "tip" the grass with your mower. DON'T shave it off. A lot of people think that the more they shave off their lawn, the better. NO! It's what it looks like when you've finished, not how much clippings you can take off.
Then of course, as I've mentioned on here before, if you don't need your lawn for anything special, plant it up! I have just completed a project of this nature for a friend. I'll post pictures of before and after when it starts performing in the summer.
Top man! It'll have to wait till I get home anyway; by which time we'll have a date at Wembley & survival is ensured!