You need to activate those versions!

The reason I asked this question is that Sage support said that I do not need to activate them with the serial number and activation code since I am simply upgrading.  He said the trial would work.  Even though I have gone through and activated these modules now I am still receiving the same error.

Could this have something to do with my issue?

"Damaged data.  Activation Log Entry. Table not found."

You need to start over.  Restore your 5.5A databases (both System and Company) from backup, and activate them in 5.6.

Do not confuse data activation (the process used to upgrade the data) with the serial number and activation codes you need to enter to "activate" the software licenses. Two totally different animals.

I made a copy of our active database and renamed it.  On my test server I launched the Sage Accpac Database setup tool and created a new company while pointing to the fresh copy that I just made.  Do I need to install 5.5A on this test server or can I install 5.6A and activate the data?  Do I now need to run 5.5A on the test server or shall I run 5.6A to activate the data?

You might get away with only 5.6, but if there are unposted batches, you'll have to add 5.5 modules to finish them.

When i go to sign in to my new test company on 5.6A I receive the message  "Program file problem. Administrative Services programs are not compatible with system data."  

Look in the CSAPP tables in both your system database and your company database.  What does the AS record show in each one?

I have edited the CSAPP table to reflect the 56A update but now I am back to receiving the following error message.

Accpac has detected module update inconsistencies that require activation. The following applications need to be installed and activated.

Bank Services 5.6A

Tax Services 5.6A

Project and Job Costing 5.6A

Accounts Receivable 5.6A

G/L Subledger Services 5.6A

The current versions of these applications as well as any others requiring them, will be disabled until they have been installed and activated.

Probably your PUs aren't up to date.  Do a Dbload of SAMINC 5.6 and compare its CSAPP records to yours.