I've backed red oak, juniper, ash, and elm with sinew... but not osage. Sinew seems to work best on short, wide bows made from light-weight wood.
Osage bows can be made narrow and I think you have enough wood there, but others with more experience with osage will be able to instruct you better than I can. Someone should chime in later...
I've heard that most people wipe the osage with acetone (to remove the natural oil on the surface) just before applying sinew.
I usually floor tiller a bow before adding sinew but I have also added sinew to a rough stave without floor tillering. However, I think the best method would be to tiller the bow enough to brace it, then remove the string and bend the limbs (in the other direction) into a slight reflex before adding the sinew. Then wait for the sinew to dry completely before final tillering. I am about to try this method on the bow I'm working on now.
The only way to accurately tell if the sinew is dry is if your bow has stopped losing weight. Even then, many bowyers wait even longer to let the sinew "cure". The curing process can take several months, depending on your patience.
Yes, you can get a bow's draw length to equal half it's length, but you have to pay attention to the set you are getting as you tiller and exercise your bow. If you are getting too much set (more that 1-1/2"), your wood is not wide enough. JMO