Thank you very much for responding. I'm going to look for a Shine Office in Boston.

Answers to your questions:

Timeline
(Dates below are approximate - yet probably all correct. I have the exact information in my files if needed.)

May 2014 - I moved so my mail was getting forwarded.
July 2014 - All of the sudden my SSDI was reduced by over $100 and I was told that was due to Medicare Part B. I received a notification letter well after the fact.
July 2014 - I called MassHealth to see if I was still covered under their insurance plan. The answer was, "yes, but only until the end of the month unless Medicare Part B is cancelled." (MassHealth did NOT warn me of the Medicare Part A scenario described in the OP.)
July 2014 - I called Medicare and asked to cancel Medicare Part B. Medicare asked me if I had other insurance and I said, "yes, MassHealth." I sent a letter confirming my cancellation of Medicare Part B. (Medicare did NOT warn me about the Medicare Part A scenario.)
end of July 2014 - I called MassHealth to make sure they knew that I cancelled Medicare Part B, pursuant to my previous advice from MassHealth. (MassHealth said they would put a note in their system so I can keep my MassHealth coverage in case the computer system is updated late. Still no warnings about Medicare Part A.)
August 2014 - I called MassHealth to make sure that I was still enrolled in their insurance plan. MassHealth said that my MassHealth plan was cancelled because their system shows that I was enrolled in Medicare Part A.
August 2014 - I called Medicare to cancel Medicare Part A too. That's when Medicare informed me that the penalty for cancelling Medicare Part A is having to pay back all SSDI money ever received and having my disability payments stop immediately. This was news to me. IMO, this is not a logical question to ask or in any way obvious.
August 2014 - I made multiple phone calls to MassHealth to see if there was anything they could do for me. (Part of the reason for "multiple" calls is that MassHealth transfers you around a lot, and often times there is eventually a recorded message saying something like, "All agents are busy. Please try again later." Then it hangs up on you.)
September 2014 - Eventually a MassHealth representative put me on MassHealth Standard and told me that that would cover all of my doctors visits even though I was also on Medicare Part A. (Medicare Part A ONLY covers overnight hospitalizations*.)
September 2014 - Since the previous MassHealth rep sounded confused and couldn't answer questions, I doubted that rep's competence. So, I called MassHealth back to confirm that I really qualified for MassHealth Standard and that my doctor's visits would be covered. MassHealth confirmed that.
October 2014 - I contacted MassHealth about a different matter. That's when the MassHealth rep stopped me and said that she saw something concerning in my file. She then informed me that MassHealth Standard will NOT cover any of my doctor appointments regardless of what I was previously told. (There's more about this in the OP above.)

The MassHealth plan I was on was BMC Healthnet through Commonwealth Care through MassHealth. (I wish it were simpler. I didn't invent this.)

I never received anything from Medicare regarding my cancellation of Part B. However, my SSDI check increased back to the pre-Medicare rate. In other words, the Part B premium is no longer being deducted from my SSDI checks. Over the phone, Medicare confirmed my cancellation because I called them.

I appreciate your letting me know that what Jen said about secondary coverage sounds wrong. Does anyone know where I can get credible information about this? I put that part in bold in the below quotation.

*I also very much appreciate your informing me that only admitted hospitalizations may be covered by Medicare Part A.