I'm sure some more experienced osage cutter will chime in, but with my limited experience, the only big problem I see is dealing with now leafy branches, new shoots with thorns and probably more undergrowth at the logging sight. I just cut a tree a couple weeks ago and wished I had done it a month earlier for the reasons listed above.
Other than that, seal the ends really well. Split it as soon as possible. Bark and sapwood is much easier to remove while the tree is still green, so I would suggest removing them within a week or so. If you remove the bark and sapwood, seal the back. I seal the ends with several coats of wood glue and seal the back with several coats of shellac.