I have noticed that once you get full on an item you get the notification that your pack is full of said item, and nothing seems to have been done with the item you tried to pick up. When you interact with the stash you can move things from the stash to your pack, but it does not appear that you can move things from your pack to your stash. I would like to build up the village but waiting for items to go to stash day by day (in-game) takes a while, unless you just sit there and sleep repeatedly. Is there a way to move things from your pack to your stash? I could just be overlooking it, or not far enough through the story perhaps.
I found a post that states you cannot, and I haven't been able to find a way to do so either. The rewards stash (from my understanding) provides you with daily resources for you to use, and that's it.
There really is no downside to carrying everything in your pack. The only downside to carrying everything is that you cannot build up a supply of items and store them anywhere (like you said in your question) since the pack has a finite amount of space per item. This really isn't so bad considering that resources are fairly abundant in the game, it just adds some grinding time to the game.