Can I fix my leaking downspout without taking it apart?
Downspouts aren't sealed because gravity prevents much water from coming back up through the lapped connections. So if that's happening something's usually plugged. 

In this case, it's because of the ice blockage. If you were to seal the system you'd just move the "leak" to the gutter, and your icicles would form there instead because the entire pipe would be full of ice. 

Eliminate the ice plug at the bottom and be happy.