Hi Justin,
For our dog Blue, we had a similar issue here in Canada - as she was from Louisiana and we adopted her in February (Brrr!!).
She would get cold and start lifting her feet and shivering and looking at us like "why are you making me dooooo this??".
So, rather than give into her, we bought her a little jacket and some boots and she loved it! She ran around like all of a sudden the cold wasn't a big deal! And... it was great for her house training too!
She still wears the jacket on really cold days - as she still hasn't adjusted completely to Canadian climate, but she rarely has to wear the boots any more.
Definitely try not to give into Gaz's shivering too often - my sister's dog learned that by shivering he could be brought inside quickly. So now he shivers if he wants to come inside before he's done anything - even in the middle of summer!
If you think Gaz is really cold, bring her in for a minute or two to quickly warm up, then take her straight back outside! Especially if you know she is due to eliminate - or she will just start waiting to go inside rather then pee in the cold outside!
And dont' make any kind of fuss of her when you bring her in, just bring her in, wait by the door for a minute with her on her leash, then take her out again.
Hope this helps!
Blue