Thanks for all of your help, everyone. I do still wish that I could combine curves (or start a new curve off of an existing one) like you can in Illustrator, though. I must admit I'm a bit disappointed with Designer in that regard.
Edit: After coming back to Illustrator for work I realize that I was both incorrect and ineffective in describing my issue. Both programs allow you to snap new curve nodes on top of nodes from other curves. Neither program automatically joins these curves, though. Both programs allow you to select both overlapping nodes and move them together, Designer just requires that you select both curves first (as opposed to having nothing selected and just dragging over both nodes with the node tool), which does feel a bit awkward, but is a non-issue. Neither lets you begin a new curve off of another curve's node – trying to do this in Illustrator defaults the pen tool to "delete anchor point" and doing so in Designer just selects the node.
Sorry for my confusion. I've really been enjoying both Designer and Photo!