Problem with heading south in June is that it's generally hotter down there.
And FWIW, I didn't do a shake-down tour before crossing the country. In fact, I only rode my bike fully loaded a week before I hopped the train to Seattle to start the trip. About 64 miles total. (Interestingly, I was nearly hit during that ride when a kid lost control of his car, clipped a telephone pole and flipped onto his roof maybe 100' behind me. Me and another motorist had to break a window with a Club lock because the car was filling up with smoke. Good thing I hadn't been a fraction slower.) The first night of the trip was the second time I set up my tent and the first time I slept in it. On the 6th day of riding we crossed the North Cascades Highway in the rain and snow. Close to 30 miles of climbing and two mountain passes then another 30+ miles to camp. Wouldn't recommend that approach, but that's the way things worked out.