Hey Tim, couple of questions:
When you say 'now have 11'' are you saying that is after the 5' addition or that you currently have 11' and are adding five more to it?
When you say you can sit the beams on the header for '3.5 inches but then they hit the roof' - to clarify, there is 3.5" of existing space since that is where the 2x4's are currently sitting on the header?
Assuming that the roof is an extension from the main home, you won't be able to 'sneak' in those oak beams without cutting back into the existing roof, the reason for that is you need the height to place those new taller oak beams on top of the header. Hanging the beams from the header will not likely work since you will now introduce a rotation force on that load bearing wall that it probably was not designed for. Assuming again that this is a total span of 11', you could double up 2x4's with 1/2" plywood in between - a common header construction practice. Assemble with construction adhesive and construction screws. That should be ample stiff to carry your snow loads in PA. As for hangers and spacing, you should consult two resources for this:
simpson strong tie for appropriate hanger sizes and a
span table. If this is to be inspected, you will have to consult with an engineer to get approval. The aforementioned sandwich is off the cuff but should work and to be conservative, frame it 12"o.c.