From what I hear, 23andMe is notoriously problematic for genealogy. People get on there trying to learn medical things, not the least bit interested in genealogy. I'd suggest uploading your AncestryDNA test to GEDmatch and MyHeritage.
It sounds like you have the disadvantage of your American roots not going very far back. DNA testing is far more common in America, and all those relations out there that might help you out by testing are overseas and not testing.
Still, sometimes you might catch a break, where, say, a cousin or 2nd cousin of your immigrant ancestors also came over.
If you've identified, via "shared matches" clusters connected to three of your grandparents, and this Odom relative isn't connected to them, that sounds pretty compelling, as to it being a relation on your mystery bio-grandfather's side. How many cMs for this "total stranger"? It really sounds like you're on the right track.
I'd try to sort out how some of those other Odom matches connect to this guy's grandmother. I'm guessing that pretty quickly a picture will emerge that will give you a great deal of confidence how you're related to these people. Brick wall smashed!
Even I (with American roots going 4 to 10+ generations back) get those DNA matches to people whose roots are entirely in the Deep South, despite my having no ancestors or known relatives being anywhere south of Baltimore. What I make of it is that there are certain DNA segments that have persisted for quite a few generations - that our common ancestor is further back than my 4th-great grandparents, most of whom I know about (the ones who aren't in Ireland, anyway!) Really it shows how practically everybody in this big country of ours are connected by blood, in at least some small way - that even we Northeasterners are CONNECTED to our Southern "brethren". From what I've seen of the history of the South, it was settled, in large part, by people from the North back in the mid 1700s. Back then, if you were looking for open land to farm (which was the only occupation back then, for the most part) your choice was to either hack your way through the forests and mountains, going West, or get on a boat and go South. There was SOME direct immigration from abroad (especially with the Scotch-Irish) but I think most immigration from overseas went to the big sea ports up North.