I rode the TransAm in 2011. We left Yorktown on June 11. We had one shower on the Blue Ridge Parkway, and a couple of hours of rain leaving Hindman KY and an hour or so of rain near Booneville KY. That was the sum total of precipitation.
Edit, we did have a couple of epic overnight storms too, but we were camped under city park pavilions so had no worries.
I packed some arm and leg warmers, neoprene overshoes, an event jacket, and a gilet. I wore my overshoes only once. After Pueblo, we had some very cold early mornings. I had a woolen base layer, and a buff to keep me warm. I mailed home my long fingered gloves. We had a couple of cold mornings in Yellowstone and in Jackson Hot Springs Montana when our tents froze.
Heat and humidity further east was a bigger issue.
I read journals of folks who left perhaps a month or so earlier than us who had dreadful weather for days on end. It's a lottery I guess. I would caution against riding busier roads in really heavy rain, purely on safety grounds. We switched on our rear blinky lights when visibility was poor.