I want to have some fun and see what it's like to run out of stickers in a battle, but I don't want to sell or toss my rare stickers. If I paste my rarer stickers on signposts in The Bafflewood, will they stay there?
Yes if you paste your stickers on the signposts they will stay there (that's pretty clever, actually!) The point of the boards is to mark the correct path through the forest; I just used the "worn-out" stickers so I didn't waste better ones. (Note: I never tried sticking a large sticker to the board so I don't know for definite that you can use those.)
Also there is a point in the game where you fall down a spider's burrow and lose all your stickers, and must pick them up again as you make your way through the level. You could wait for that if you like.
Aside from that, there is no way to store or save stickers. But do remember that the majority of stickers can be bought from shops so if you have plenty of coins (which is likely) you can fill up your album easily. For Thing stickers you can buy the Thing again and convert it to a sticker.