Can't see old backups on Time Machine

Question

Originally, I noticed the issue when I went to do a backup and it failed because there was not enough space on the disk. Then I went to see if I could delete old backups.  In Time Machine it shows today and one other backup, 6 days ago.  It should be doing backups regularly.  I have had the iMac and my external drive for years.  I did update to Mavericks OS several months ago.  That is the only change I can think of.  I have seen older backups and used them before now.  Not sure why I can't see them anymore and why it says my disk is full.  Why isn't it deleting the old backups if it's full?
So I guess I have three main questions:
1 Why can't I see the old backups?
2 Why aren't the old files being deleted if it's full?
3 How can it be full if there is only one backup?
Thanks

Answer 1

These instructions must be carried out as an administrator. If you have only one user account, you are the administrator.
Launch the Console application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad. Click Utilities, then Console in the icon grid.
The title of the Console window should be All Messages. If it isn't, select
          SYSTEM LOG QUERIES ▹ All Messages
from the log list on the left. If you don't see that list, select
          View ▹ Show Log List
from the menu bar at the top of the screen.
In the top right corner of the Console window, there's a search box labeled Filter. Initially the words "String Matching" are shown in that box. Enter the word "Starting" (without the quotes.) You should now see log messages with the words "Starting * backup," where * represents any of the words "automatic," "manual," or "standard."
Each message in the log begins with the date and time when it was entered. Note the timestamp of the last "Starting" message that corresponds to the beginning of an an abnormal backup. Now
CLEAR THE WORD "Starting" FROM THE TEXT FIELD
so that all messages are showing, and scroll back in the log to the time you noted. Select the messages timestamped from then until the end of the backup, or the end of the log if that's not clear. Copy them to the Clipboard by pressing the key combination command-C. Paste into a reply to this message by pressing command-V.
☞ If all you see are messages that contain the word "Starting," you didn't clear the text field.
If there are runs of repeated messages, post only one example of each. Don't post many repetitions of the same message.
☞ The log contains a vast amount of information, almost all of which is irrelevant to solving any particular problem. When posting a log extract, be selective. Don't post more than is requested.
Please don't indiscriminately dump thousands of lines from the log into this discussion.
Please don't post screenshots of log messages—post the text.
☞ Some private information, such as your name, may appear in the log. Anonymize before posting.