Condolences, Holy Roller. With a journey like this, you'll have a lot of time to think, and I hope it takes you to a higher level.
You'll find a cross-country cycling trip is about the people -- the ones you meet, the ones who approach you out of interest and curiosity, the ones you join on the road. I think if you are open to why you're doing this trip, you'll find that most people go through an identical catharsis, some younger, some older. We generally don't talk about these turns in our lives... You'll have the opportunity to share their stories.
5 or 6 weeks -- that's just about enough time to dig deep into the journals on
crazyguyonabike (CGOAB). Most of the questions (fitness, gear, routes, etc.) are answered in there, but the stories will rope you in. On that site, search for "TransAm" or "Western Express".
The logical first choice for your trip is the
Adventure Cycling (ACA) TransAmerica and Western Express routes. The comment above about "most people going W-E" isn't accurate, it's about half-half. If you start in mid-May (even a little earlier if possible), it's ideal for going E-W. The thing to watch out for is -- if you take the Western Express route across Utah/Nevada -- try to plan on that before mid-July. Any year can present different weather conditions, but that will give you a good chance of not expiring in the desert heat...
Another choice is to take the TransAm to Oregon and ride down the coast -- obviously more miles and time. But you'd avoid the long desert stretches.
The benefits of going E-W with a start in April or May is to escape the east coast heat and humidity of mid-summer. For a complete TransAm, generally aim to cross the Rockies (Colorado-Wyoming-Montana) in mid-July, no matter which direction you're going. Your plan fits in to this (starting on the west coast, you're looking at mid-June). Nevertheless, everybody has schedules and reasons for deviating a bit.
Look over the ACA and CGOAB sites. And this one. Lots of good info to get you started.
-- Mark