That is a bug for the first one. Ill get that fixed in the next update.
not sure about bug two ill investigate that.
you need to use an ember heart at the saw mill and complete the quest to fix the sawmill. I think i could have better popups there to make it more obvious. you will be one ember short from that chest being missing it shouldnt have aloud you to collect it if your full. but you only need 3 for the demo. I would advice not using an ember on the pine forest camp to save you needing to restart.
I have updated with a new build with a few Qality of life changes chests can no longer be looted if your inventory is full or you cant carry the item. sawyer is now there by default so you dont need an ember heart to do his quest. mushrooms in the mines now keep track of there state and are saved. the UI text is now all using the same font. will hopefully get an update tomorrow with a fix to the sawmill and forge not saving items that are left in them. hopefully this helps and you wont have to start over :)