Google+ is confusing ...

Let's break it down
"My Google + situation is strange. I run my own business and started a Google+ account for it a year or two ago."

A: There is no such thing as a "Google+ Account".
B: Never start a Google account for a business.
"I'm not sure how this happened, but I seem to have two different pages that are sometimes the same, sometimes different."
With that "account" which you set up for your business you set up a Google+ Page, which is a Brand Account. That is a "business account". So you have a standard account with the business name and a brand account with the business name.

As a person what you should have is Your personal account, with your name, which "owns" the Brand Account (Google+ Brand Profile), with the Brand name.
"If I go to Google+ by logging in at Google, it takes me to a page which shows all of my posts roughly over the last year (as well as other people's posts)."
That is the "Home stream" for the standard account profile. That contains posts from profiles you have followed (or circled). You have probably also followed the Brand Profile, so it also contains posts your Brand account profile has posted.
"If I access Google+ from my business page and then look at my posts, it shows me only mine, and the posts stop at around August of 2016."
Then you are (probably) going to the Brand account profile, which shows only posts the brand profile has posted.
"All of my posts since then are not there, yet it shares the same posts with the above mentioned page if they were made previous to August."
You may be looking at the Brand account profile as the owning account profile and may not have posted earlier posts to public. Either way here you are making assumptions that are most likely wrong.
"My ranking has been dropping, and I'm wondering if it's partly because Google doesn't recognize the posts over the last 6 months as being associated with my business."
Unrelated, or a different kettle of fish. What you first need to do is understand the difference between the Brand Profile and the Profile on the account you created "for business", which was the basic mistake. It's a very common mistake.
"I would like to have just one Google+ business account. Does anyone know how I can do this and why I might be having these two "sort of" different pages?"
Understanding the difference between account types is critical and key to achieving what you want.
One Google account. With that one Google account you manage your Brand Account or accounts, which represent your business.

That is the position you and most business owners would want to get to.

YOUR personal Google Account should "own" the Brand Account. Which is the identity of your business online.

There is no "login" for Brand Accounts. 

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