When I look at June, the Reds games on June 23, 24, and 25, come up as being available.
The blackout dates they posted are 4/3, 4/29, 5/12, 5/27, 5/28, 6/10, 6/26, 6/27, 6/28, 6/29, 7/4, 8/12, 8/26.
On the filter at the top of the eligible dates, you can select All Games or display them by the month.
Also, since you can only exchange tickets at the same ticket price or at a lesser value, it's possible that there are no seats available in your section.
It doesn't look like you can combine tickets from multiple dates, you just have to exchange them one game at a time, but since you can select your seats you should be able to exchange seats from two different dates for a single game with the seats together as long as they are available.