Redundant Core - Spanning Tree Design Questions

I recently moved into a networking position managing a large existing network. I'm more than a little fuzzy on spanning tree, and the organization has had major problems in the past that were attributed to it, so everyone is afraid of it and doesn't know anything about it besides: don't touch it.

We have redundant core switches with over a hundred buildings connected to both. The core switches also serve as our core routers, but that seems to be working fine for now. In my poking around I've discovered that Core A has had it's priority set so that it is the root bridge, which seems normal. However, Core B seems to have a priority that's made it even lower in the tree than most, but not all, of the head end switches. This results in many of the Core B ports being in the blocking state, rather than the designated state.

This doesn't make sense to me. It seems to me that I'd want B to be right behind A in terms of priority, so that if A falls over then B becomes the new root. That would mean each headend would be responsible for putting one of its uplinks into the blocking state. Additionally, there doesn't seem to be an L2 connection between A and B, so B has to get to A via one of the building headends, and that building has all the VLANs tagged across it.

I've got a diagram of what things look like now: https://i.imgur.com/05sFd3w.png

My question is... what should this look like? Should I modify the priority on B so that its ports are in the designated state instead of blocking? Should I stand up a new L2 link between A and B for all the VLANs so B can get directly to A?

Thanks! -ljb2of3