Dash and other electrical problems after alternator/battery replacement

A few weeks ago, I had to get the battery and alternator replaced.  Since
then, some problems have developed.  The battery light comes on some of
the time, even though the alternator and battery show full charge when
tested.  I regularly hear a clicking sound underneath the steering wheel/dash.  
If the cab light is on, it flickers on and off with the clicking.  Sometimes, the
dash will show the back hatch as being open, but then I'll hear the locks click
and the light turns off.  If I go above 60 mph or so, the clicking goes pretty
fast and then the whole dash simply turns off (no mileage, no speedometer,
no ability to turn on turn signals or wipers, also the electronic locks won't
work).  Once I brake and get back down to 40mph or so, the clicking starts
again and the dash turns on.  I've also noticed that the car idles unevenly
and sometimes after the clicking, the AC will be blowing higher than it was
before.  If windshield wipers are going, their speed fluctuates if the clicking is
going on.  Mechanic said clicking is probably coming from "relays."

This electrical stuff does NOT affect the A/C or radio.

Any idea what the heck is going on?