I have checked the AutoMerge Editing box in the Mesh menu, and dragged faces so they are as closely lined up as I can make them, and clicked Remove Doubles in order to adjust the merge distance to wider. Still I can almost never get things to merge. Is there something I'm missing?
The default automerge distance is very small, as it's usually used with Vertex Snapping.
To use vertex snapping, set the snapping mode to Vertex (⎈ Ctrl⇧ Shift↹ Tab):
Then hold ⎈ Ctrl while transforming vertices (or toggle the magnet icon in 3D view > Header)
Now automerge should work as expected:
To increase the automerge distance, adjust the Double Threshold in 3D view > Tool Shelf (T) > Option > Mesh Options: