I tested the relays to see if they clicked when operating the switch. They both did, but the trim/tilt motor didn't respond. Then I opened the "UP" relay to jumper directly to the trim/tilt motor wire from the battery to see if that would work. To my delightful surprise, the motor raised as it is supposed to. I was very excited at this, so I opened the "DOWN" relay to jumper directly to it and it didn't work. I thought a little about it and put the "UP" relay back together and tried again. Now the motor went down as it is supposed to. I was very happy to learn that I didn't need to replace the trim/tilt motor, but I wasn't very happy that I would have to jumper directly to those wires to operate it. So I put everything back together and again to my surprise the trim/tilt motor is working perfectly normally (for now). I'm going to replace both relays now, thinking that they are dying a slow death.
Thanks Swagonmaster for all the help, let me know if I've come to the wrong conclusion.