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The "best" time depends on several factors, i.e. temperature, rain, wind direction, date that lodging/campgrounds open for the season, etc. Since "best" is highly subjective for some areas (especially weather), I would suggest you check out WeatherSpark.com. They have an excellent climate database that shows the averages for a lot of the world. Also, if you are camping, be sure to check a few campgrounds as to when they open.
You don't say how many miles a day you plan to do and be sure to factor that in. If you do 40 miles per day, you won't have to worry about when campgrounds open west of the Mississippi but if you plan on 100 per day, then yes, you will need to factor that in.
Personally, early April is a bit early for me but I like warmer temps.
Finally, if you look at the Route you want to ride
https://www.adventurecycling.org/routes-and-maps/interactive-network-map/ then click on the "Logistic" tab for that route, ACA gives a broad idea of the expected climate.
As a side note, the Eastern Express does not go to New Jersey but the Chicago to NYC route does. I wasn't sure if you plan on riding down to Virgina via the Atlantic Coast or DelMarVa route and then starting the TransAm or riding west via the CNYC route which I assume is the one since you mention the Eastern Express. Regardless, the info above is the same.
Tailwinds, John