06 Buick rainier cxl differential locking up

I have an awd 06 Buick Rainer (cxl) that I
recently replaced a back tire on. The back
differential is locking up on it, and I'm being told
this is the reason why. I also have other misfire
and broken bolt issues with it as well, but I don't
want to pay $3300 to repair these, if this is not
the issue with the back end, and have myself in
a world of debt fixing the differential issue after
this. I also had a stabilitrack warning light on, as
well.  
The garage is 4 hrs away, so I'm basically doing
this over the phone. Would one new tire cause
this? Should the garage be able to tell why it's
locking up without replacing the tires first (they
wanted me to pay $1200 for a new set of tires,
as well, but I have a set, and wondering if they
were just trying to push the tire sale on me
before actually diagnosing the problem first)?
The vehicle is a rebuild, I've had no other probs
with it up until now, and it says 300,000+ *miles*
on it, but I'm assuming the motor has been
replaced.
The price of vehicles up here is ridiculous ($7-
9000 used pos), so paying $3300 for repairs is
reasonable considering the alternative. It ran
great before I drove it hard for 4000 kms across
BC, and then 700 kms a month after this for 4
months, and with no garage here, I had to tow it
out when broke down.
I love the vehicle, but if I'm going to need to put
more money in after this, I'm going to have too
much invested, whereas now I can still back
away. I hope u understand what I'm trying to
explain, and any advise would be greatly
appreciated