2x4's perpendicular to the existing 2x12 ceiling joists, at 16"o.c., hung with joist hangers. NOT 2x12's....they are too heavy.
You need about (40) 2x4x8's.
And you can't make a longer 2x12 beam from shorter ones without a lot of difficulty.
The method outlined above to install short 2x12's does nothing but add the loads of the new 2x12's to the existing ones. And you still need additional blocking at 16"o.c.
Save them for some other use.
As far as a lot of joist hangers goes, you have two solutions. Either use a combination of toe nailing and end nailing or a combination of joist hangers and end nailing,
And you can not use the space above for storage, period... It's insulated and you can't put down any decking. It wouldn't support the extra loads anyway.
Nailing patterns:
Single nailed:
Double nailed: