I installed a new garbage disposer okay, but the pipes don't connect precisely. As you can see from the photo, if I rotate the U-shaped trap pipe toward the garbage disposer out-flow pipe, it connects fine. The problem when I do that is that it unscrews the U-shaped trap pipe a little from its outbound connection, so that it drips a little bit from that connection. Do I have to cut and make new pipe connections, or can I just shut off the source of the drip with some kind of plumbing sealant? I hope it is the latter.
Thanks for your time.

Get a drain extension, cut to length.

While you are at the store get a new J bend. That one looks like its ready to start leaking any time.

Purchase the plastic compression extension tailpiece shown in Harolds image. Cut the pipe from the disposal back a bit and cut the tailpiece to fit. If the bottom of the "J" bend feels spongy replace it at the same time you replace the tailpiece. Good luck, Tom