When does the game determine which fossils are which

There are three common symbols on your DIY cards:

• Hammer: You have the materials to craft this DIY at a workbench
• Basket: You have one (or more) of these items in your pockets
• Checkmark: You have crafted this recipe at least once (Which is relevant for certain Nookmile achievements, I believe)

Additionally, the recipes will have backgrounds that match the general kind of materials you need to make them. For example:

• Green: Plants / weeds
• Brown: Wood
• Grey: Iron Nugget
• White: Other (e.g. bait / medicine)

I've only seen being Full affect two things:

• Hitting a rock with an axe will consume 1 unit of fullness to break the rock. (This is not ideal if you're trying to farm the rock for resources)

• Using a shovel on a tree will consume 1 unit of fullness to dig up the entire tree, placing the whole thing in your inventory. It can then be replanted, fully grown.

So if you're not trying to do some landscaping, I haven't seen any benefit to eating at all.