Hey numbersguy, It sounds like you got it in one! Since the sound of the pump gets louder when the water has been pumped out, yes you are dry pumping! That should not be a problem with this type of pump. As it is far better to have a little dry pumping rather than not enough pumping time and have dirty water left to mix with the rinse water for the next cycle, it might be wise to continue to keep the screen and filter clean. If you notice the hum and vibration becoming more pronounced, look into buying a pump now, so that you will have it on hand for the replacement project that will follow.
As far as gettin to the pump, well, you might have to take the unit out from under the counter to be able to do the job easier, rather than working blind under the silly thing. I wouldn't recommend cutting a hole in the basement ceiling just to change the pump! Scope out the cost of your parts, and weigh that against the cost of a new machine. You might be surprised! Good Luck!