Yes, I've done fall cross-country tours a couple of times.
Departing from the Washington coast in early August is ideal.
You will have pleasant weather through the West.
When you get to the Plains, you should have a reprieve from summer's heat.
And you may catch the beginning of fall in the East.
(But fall is coming later nowadays, so probably not.)
I'm not sure what route you are planning but here are a few possibilities.
If you can - start out at the very northwest tip of Washington.
You're going to have plenty of time if you have three months.
The Makah Reservation has an incredible museum with artifacts from an ancient village that was uncovered by a mudslide.
You can camp at Shi Shi Beach and hike along the coast for a day.
There is bus service out to Neah Bay and they take bikes.
If that's not possible, then definitely do the San Juan Islands. There is a great campsite on the west side of San Juan Islanf (Friday Harbor) but this is the busiest island and Friday Harbor can be a real zoo. I recommend Lopez Island and Spencer Spit State Park - lovely, serene riding - great way to start a trip.
Figure you will be doing the Northern Tier part of the way, at least. It's a great route across Washington. In Idaho you may want to consider heading north to Bonners Ferry so you can do the Yaak Valley in far western Montana. I think the NT used to route you this way but they changed it. Why, I don't know. But the Yaak makes for some great riding too. There used to be a little country store at the highway junction in the Upper Yaak.
You absolutely must do the Going to the Sun Road and give yourself a few days in Glacier. It's certainly worth heading up to Many Glacier - an incredible region with great camping and day hikes. Then I would take US 89 thru Great Falls all the way down to Yellowstone. US 89 has low traffic north of Great Falls and great views of the Front Range. There's a beautiful riverside trail along the Missouri in Great Falls. Then US 89 is incredible once you turn south at Belt. (And I've been between Glacier and Yellowstone umpteen different ways.) You can access Livingston north of the river without having to get on I-90. Then there's a back road on the east side of the river into Yellowstone.
If you take your time you'll be getting into Yellowstone in mid to late August and the crowds will have thinned out considerably. I'm not sure if you plan on taking the TransAm from Yellowstone of heading east. US 16 has some spendid scenery across northern Wyoming (even better is to head out the northeast entrance via Cooke City and take the Chief Joseph Highway) - - Cody, the Bighorn Mountains, Buffalo, and Devil's Tower - then you can hit the Black Hills - there's a super bike trail - the Mickleson(sp?).
Anyhoo - my guess is that you would be thru the Great Plains by mid Sept at the latest - and I think you would probably reach Connecticut before the end of Oct - probably mid-Oct. Almost any back road across South Dakota is nice. I recommend Hww 44 - do the Badlands Loop - then thru White River. Neb 12 is lovely thru Keyapaya and Niobrara country. Iowa has great maps at their DOT site showing traffic and suitability of county roads - use county roads! - plus there are numerous rail trails - especially the one into Dubuque!! There a great trail along the Illinois and Mississippi Canal tow path across Illinois. Zero cars!
There some nice bike routes in northern Penna - check out the Penn DOT site - they have maps. Maybe the Catskills then come into Conn in the northwest corner near Salisbury - - not sure where you plan to end up BUT you should definitely find water's edge. Block Island can be lovely in the fall - but the New London ferry is summer only - - you'd have to go into Rhode Island.
Best - J
PS - Make sure you have a good flashlight and batteries since the nights get longer and longer and the mornings get cooler and cooler. I always do lots of reading on my fall trips.