The beav is a great little river, while it is dam controlled it can vary a lot. It can go from 1000 cfs to 300 overnight or vise-versa. The upper stretch does not open for fishing until the 3rd Saturday in May and as such will be very crowded for the first few weeks of the season. Plus it gets over concentrated when surrounding rivers are dirty - makes for wall to wall fisher-folk above Grasshoper Creek. I've not floated it below that (but need to to finish of my runs of all floatable water on the three-forks of the Mo).
The Bighole is great. Not so sure about run-off not starting for a month, it was at 5K friday (5 feet), only a foot off flood stage. It's dropped down to the low 3K's as I write this. It's usually at or above flood stage during salmon fly's and as Sembob says it looks dirty from a distance but usually has a lot of vis when you really look at it, even when flooding. The canyon (divide to Melrose is a great float, fun, beautiful and there is camping along the gravel roads that parallel the river (south side). Another cool float is Maiden to Browns, you can take the right channel at Melrose which is fished a lot less than other stretches in the area... There are also a campgrounds at the divide bridge and Melrose (both boat ramps). I also really like some of the upper stretches, East bank to Jerry Creek is fun and if it's up you could stretch it to Dewey. Nice camping on the south side of the river above Eastbank and Eastbank it's self isn't a bad campground. It's also nice below Glenn, not as many people but a more open braided river, totally different character than above but the roads can be a bear then depending on precip.