Getting from the N. Jefferson trailhead to Jeff City is do-able, but you will be next to highway traffic. There's a shoulder on the bridge that cyclists use all the time. It might not be pleasant, if you don't like riding next to traffic, but it can be done.
Washington is a bit more difficult. The highway shoulders from Dutzow to Washington are not suited for bikes, and the Missouri River bridge is only wide enough for one car in each direction. An experienced cyclist could do it, but it's not recommended.
One alternative is to use a
shuttle service. Mainly they shuttle riders from one point to another along the trail. But if you call ahead of time, you might be able to find one to take you from trailhead to train station.
Another option: some of the B&Bs in Washington and Jeff City (and Hermann too) offer a ride for their guests from the trail to the inn. So if you're going to overnight in one or both towns, ask your innkeeper.
Good luck, and have fun!