Generally, early fall will be the best time to rennovate your lawn. If you wait until late September to mid October you won't be dealing with broad leafed weed activity.
In order to grow a beautiful lawn, you need a deep root system; anywhere from a minimum of 6" and beyond is ideal. If you don't have an ideal seed bed, you are waisting your money. Unless you already know the soil makeup of your lawn, you'll have to take a core sample here and there to see the content of your existing organic matter.
A galvanized pipe cut at a steep angle on one end will serve as a corer. I'd make one about 24", drive it straight down with a sledge, and take about a 12" sample. I'd do that in a number of places through out the lawn area.
If you don't have lots of organic matter in the root zone; you'll be set up for failure, as the micro organisims have nothing to feed on. Since your soil is quite rocky, you have excellent drainage, and what I call free soil. Meaning water percolates through at a faster rate than in a clay soil. A good medium for root growth is like a sponge that absorbs quickly enough to prevent run off on slopes, yet allows the water to pass through the root zone, down to the subsoil.
Here's what I might do. Unless you want to kill the existing lawn, including the roots with glyphosate; and if you don't mind the neighbors pointing at your place; stop watering, and let it turn brown. You now have a nice mat of dying organic matter. Only watering will revive it, but we wan't to wait until after performing a light thatching.
Now you are ready to reseed, fertilize, and water. The hot summer days are done, and night time temps have mellowed. You may want to add straw, but you shouldn't have to.
Whatever you do, don't rototill a live lawn. You will bury live root clumps at varying depths throught the root zone. Since they won't have decomposed yet, when the new rootlets hit the root clumps, they will stop on their way down to the proper depth for continual good growth.
You could go with your original plan. Just give the lawn a decent raking to create spaces for the seed to nestle in, and germinate. Rather than straw, if you aren't dealing with a huge area; top dressing with a combination of steer manure/mulch, can work wonders. Rental yards have screened spreaders about the size of a lawn roller. Easy, and quick to get the job done. Once you've top dressed, water as if you are starting a brand new lawn. If you do it now, you may create a lot of weed growth.
If I were doing it here in Southern Oregon, I'd wait until fall. I don't have a clue what things are like in Northwest CT.