Voltage gauge moving often

Had a dead jeep.  Charged battery overnigt with
trickle charger, checked the battery with
multimeter in am, indicated battery was crap
(12.22volts before starting and 11.93 after- when it
should be higher like 13 when running).  Took to
Canadian tire where they used their handheld
machine and the  load tested it   it was right on
cusp but parts lady said it was good.   

Came home and started jeep and tested alt by
pulling of battery terminals.  Couldnt remember if u
take off ground or positive so did both.  Jeep died
both times. So changed out alternator.  Now I have
squealing even tho belt its fine for wear and its
tight (but it is -20right now), along with voltage
gauge going from 13 to 15.  What could this be?  

I took vehicle to part source and they did running
diagnostics.  It said my battery was crap.  We
unhooked and put it on full machine (not hand
tested) and it said it was needing replacement with
395/695 for juice (but day before parts lady load
tested and said it was good despite my multimeter
reading otherwise-ugg). So can a crapped battery
cause this up and down on volt gauge on dash? Or
is it something else? If so, what could it be?  How
do I test for serpetine belt slipping....its in the
grooves and appears on properly and its tight, so
dunno.  It is cold but it was cold before and not
squealing.

Thanks in advance