In NJ, from Labor Day through early Oct is the window for seeding.
What you're looking for is cooler and wetter weather, so that you don't
have to constantly apply water. Competition from weeds declines with
lower temps too. However, it has been so hot and dry this year, that
unless you can easily and economically provide a lot of water, this
year has been very poor for seeding.

As for watering, you need to keep the seed constantly moist. In cooler
weather, that could be a 15 min watering 3 times a day. If it's in the
80's, then it may require watering every few hours during the daytime,
which is why this year has been so poor for seeding. If you need to do
it, I'd be looking to do it no later than the next 2 weeks, hopefully
right before forecasted rain. And if you can't water it, then I'd wait
until spring.

The extra water for a few weeks this time of year shouldn't harm the
existing grass. The advice to water deeply once a week is to encourage
deep root growth. And that is predicated on having good deep topsoil.
If you don't than you may need to water every 4 days or so. Keeping
the grass constantly wet is not ideal for the existing grass, but with
cooler temps, the chance of problems like fungus are diminished.