After doing some research on my Johnsongrass infestation I've come to the conclusion that I was on the right track when I mentioned mowing it.
Accordding to what I've read, Roundup is the only herbicide that will get rid of it for good. Countinous grazing will weaken a stand to the point that it will die out, but that isn't an option. Frequent mowing will have the same effect, but that also means knocking my other grass down. I'm going to try to split the difference and mow it high and cut the tops up. Hopefully that will keep it small and it won't be as stemmy as it was last season.
Of course, since it's never going to stop raining, I doubt it will matter.
On a totally different note; a couple of weeks ago I mentioned my buddy Chuck. I may or may not have said anything about it at the time, but his daddy was a circuit-riding Methodist preacher in Texas back in the twenties and thirties. Chuck told me he once preached a sermon about "...the Baptists and the Johnsongrass are a'takin over East Texas..."