"Anyhow back to drags (and not the queen type). Do folks here use dry or green sticks when using a small diameter, but fairly long stick to get hung up in grass and brush?"
I have used green conifers to act as both blocking and a drag,for cats,in forest.Cats don't usually take a drag very far.Most of our grassland has been grazed down by cattle by the time season rolls in so any drag would have to be very aggressive,very heavy or both.There is little to snag on so weight hopefully will slow an animal way down and tire it in the meantime.
I have used dry poles but the bottom line is,it must be plenty solid for the targeted animal.Green can be much stronger but when trapping on public land,I can't cut green.On private,I have permission to.Most of my clogs,when used need to be heavier as there is always a chance of a lion encounter in forest country and some possibility of contact with wolves in the prairie.
I can tell you of one tree that grows here that is super tough,green or dead.That is the mountain mahoghany.It is a bushy tree with very tough,scratchy limbs and bark and extremely tough,sinewy wood.Wood workers say it destroys woodworking tools.I know I snapped off two lag screws,trying to hang a bear box for a trail camera.I gave up.You can't find a straight limb or trunk on one,so no poles,but the bushy tree might make a great drag as is.A standing tree is solid to anchor to but there could be a potential for fur damage.
Another tree here that has the potential to remain strong after becoming dry,is the juniper.The ranchers tell me juniper fence posts can last in the ground for 30 years.I have access to some old juniper fence posts.Some are still solid,some not.
In the lower 48 there is such a variety of topography,tree types and annual rainfall,some areas may have very little dry stuff that can be trusted.It is a good question and I would like to hear from others in different habitats.