Separating two blogs on one site

I have one site, but I want two separate blogs on the site.
I want each to have its own archive, template, etc., so that there is no overlap between the two blogs.

Right now, I have each blog’s posts set to their own categories to separate them.

For Blog A (primary), everything is good to go.

For Blog B, I’ve created a unique category page, complete with unique widgets and template. This page is good to go as well.

From here, there are two problems which I believe have similar solutions:

Problem #1 is that when I click on an individual post on Blog B’s page, it reverts back to Blog A’s template (including a “previous/next post” link to a post in Blog A). What I want here is to go to a page using Blog B’s template, as well as the “previous/next” links only linking to Blog B’s posts).

Problem # 2 is that when I click on an archive month on Blog B’s page, it just reloads the same URL rather than going to a new page where it is archived by that month.

Blog A example:
http://davidfranklin.net/blog/

Blog B example:
http://davidfranklin.net/blog/category/dare-of-the-day/

Thanks!