What are my options when trying to adhere felt to a 3D crocheted surface?
That sounds like a perfect use case for needle felting. Instead of using ready-made sheets of felt, use the loose fibers of rowing or batting to create the felted face right on the spot. 

With a bit of practise you can achieve colour effects similar to painting by layering and mixing different colours, like the blushing cheeks in your turnips.

Instructions abound on the net. Felt Alive's seems pretty exhaustive, covering many aspects and with ambitious projects.

LilBlueBoo.com shows how to make a face on a base and Craftsy.com has an image on a knit base, similar to your project: