archiving old posts

Which widget?

usually the widget or the page you can configure how many posts to display and the rest are found with a button at the bottom of the page indicating old posts.

This is the site.
https://sheboyganseniorcommunity.com/rehab-therapy-sheboygan-wisconsin/

Widget is called Primary Sidebar.  
http://screencast.com/t/bMldh57t

Within that widget is a "sub widget" called Recent Posts.  I can dictate how many Posts to display but I don't see where to set it to show a link to archived ones.

Recent Posts won't show archives.  There is an Archives Widget that does...I would just add that below your recent posts.

Yes, I understand that the widget only shows recent posts.  We told the widget to display the 6 most recent posts.  When I added the archive widget, it displayed months rather than post names and the months had the SAME posts as were listed under the recent posts widget.

We don't want to necessarily have archived posts as a widget since I don't want the sidebar to be that long.  I was hoping for a solution that we could possibly put the widgets that are OLDER than the last 6 that were entered onto a new page.  Is that even possible?

Here is screen shot when I had put the archived post widget live.  
http://screencast.com/t/SnrVVk2cyd1T

Each of the months listed correspond to the recent posts listed above.

When I added the archive widget, it displayed months rather than post names and the months had the SAME posts as were listed under the recent posts widget.

Yes, that's what it does.

I was hoping for a solution that we could possibly put the widgets that are OLDER than the last 6 that were entered onto a new page.  Is that even possible?

Still not 100% sure what you are looking to do, but I think I have the gist of it.

What it sounds like you want is a widget that can run an offset.  So the most recent 6 posts appear in one widget and posts 7-whatever appear in another?  If so, then look into this plugin:

https://wordpress.org/plugins/flexible-posts-widget/

We don't necessarily need a widget although it might have to be what we use.

Is it possible for old posts (older than the past 6) can be all listed on one page with links so you can click on the ONE page which has ALL old posts and within that page, click on the one post TITLE that is of interest to you.  That way all older posts are on ONE page all together.