There is a winding between transformer T1 and T2, that's a wire. So you will get continuity.
First check by touching your meter leads together, that is considered zero. Then try between T1 and T2. I bet it is higher. That is the resistance of your transformer primary winding.
So, you are still blowing fuses. Hmmmm.... If you shorted what you said you did it should have only blown the fuse and left everything downstream, all that important stuff, alone.
I guess I'm stumped. And I will be proven wrong.