Is there any way to find the mushrooms faster and/or make them show up faster?

No there isn't. Sorry. Just have to keep a watchful eye out for them. Some people claim they come in groups of 2 or 3, one rt after the other and then none for awhile. If you're needing food and want to use the "mushroom trick", mushrooms will show up randomly even when the game is on pause. At that point you can put evry child in your village on the mushroom and each of them will gather one. (remember to unpause so they will carry them to the food bin)