This.
And we’ll send you over some excess 'r’s.
I answered yes, but much of the time I’m not, because I am emphasizing the word, because otherwise I’d use the contraction “I’m”.
But if I were to just casually use the phrase “I am” it would sound the same as “I M”. However, if I were to use it after a consonant for some reason (for instance, “I am pleased that you are early, but am displeased that you are not prepared”), it would be distinguishable from “M”.
When I say “I am icy” and “I. M. I. C.”, they sound mostly the same (but not quite identical). But if I read the thread title out loud, they sound different. Likewise, “Pan Am” and “Pan M” sound different.
Based on the “aygs”, I’d guess Mississippi or northeast/east central Louisiana.
This is how I pronounce them.
I declined to vote, because for me sometimes I pronounce them the same and sometimes I don’t.
In isolation: Different
In connected speech, but with emphasis: Different
In connected, conversational speech: Normally the same, although “am” can get reduced even further (to “schwa+m” or even a tack-on “-m”) depending on the phonetic context.
In my dialect, the contraction “I’m”, at conversational speed, pretty much rhymes with American “Tom”, “prom”, or “Mom”.
Ronnie Corbett: L.O.
Ronnie Barker: L.O.
Ronnie Corbett: R.U.B.C?
Ronnie Barker: S.V.R.B.C.
Various Characters: [enter waitress carrying a large ham on a silver salver] L.O.
Ronnie Barker: L.O.
Ronnie Corbett: L.O.
[exit waitress into kitchen]
Ronnie Corbett: F.U.N.E.X?
Ronnie Barker: S.V.F.X.
Ronnie Corbett: F.U.N.E.M?
Ronnie Barker: 9.
Ronnie Corbett: I.F.C.D.M.
Ronnie Barker: [insistently] V.F.N.10.E.M.
Various Characters: [enter waitress from kitchen still carrying the ham] A. V.F.M.
Ronnie Corbett: R.
Ronnie Corbett: O.
Various Characters: C. D.M.
[exit waitress into kitchen]
Ronnie Barker: O.S. V.F.M.
Ronnie Corbett: O.K. M.N.X.
Ronnie Barker: M.N.X.
Ronnie Corbett: F.U.N.E.T?
Ronnie Barker: 1 T.
Ronnie Corbett: 1 T.
Ronnie Barker: O.K. M.X.N.T.
[he finishes writing it on his pad with a flourish and calls at the kitchen door]
Ronnie Barker: M.X.N.T.4.1.
Different for sure. But for me, “am” does not rhyme with “Sam” either. I don’t know the phonetic alphabet, but for me, “Sam” has a tense vowel (as does “man”) while "am has a lax vowel (as does “ran”). Any phonetician would conclude that I am from Philadelphia, which I am.
No, you are definitely from Helicon. Don’t play these mind games with us, man.
For me as well.
I’m from the same general area as the OP, northern Virginia, just outside of DC.
Am I? = ham
I am = hem
We have plenty of Rs over here, we just like to confuse you by taking them off the end of our cars and attaching them to new ideas. 
I’ll be darned. I checked “different”, then realized maybe I do say “em” for “am” a lot of the time.
I was directed toward this realization first by carefully remembering how John Lennon sings it in the first verse of **I Am the Walrus (“I am he and you are we…”); whereas, in the eponymous refrain, he does pronounce it “am”, presumably because it’s enunciated more emphatically there.
Well, the time when it came up was the other way around. Last year I was living in dorms, and my RA would call up the stairs to try and get people to come out for intramural sports: “IM soccer!” “IM volleyball!” I would always say back “No you’re not!” My friend was completely confused about why I would do this, because to her it didn’t sound like our RA was saying “I am” at all.
Looks like most people voting in the poll wouldn’t have known what you were getting at either. Personally I would have guessed you were associating the IM with instant messaging sooner than I am.
Oh, she was there - what wordplay I was trying to come up with wasn’t the issue. We lived in the same dorm and had perfectly compatible knowledge of what IM sports are. It was only after she said it didn’t sound like our RA had said “I am soccer” that we started having this conversation.
But I did figure that going through the whole story would just confuse people who were trying to answer the poll.
I’m from this area as well, and it’s rare that I hear anyone say “M” and “am” so that they sound the same.