I figured out what was wrong with my machine, from the FAQs on the official Singer website if you're curious. Apparently all that loopy thread on the back of the fabric is called "birdnesting," and it happens when there's not enough tension in the needle thread, which is generally caused by mis-threading the needle. I reloaded the bobbin at least three times, but I did not think to make sure the needle thread was loaded correctly. Um, that's possibly not the correct terminology. Anyway! Now I have another set of coasters I'll never use! This set is definitely the best yet. They're all basically the same size, and you can barely tell where I left a gap to turn them.
(Are you into coasters? Would red/blue or pink/brown match your decor? Drop me a line, you could be the proud owner of a new set of handmade coasters.)