My grandfather was a son of a bitch and had poor sight. Cataracts, you know.
Parody's aside, why is the title of this thread 'my great-grandmother' but the story starts with 'my grandmother'?
If she was your grandmother and she died in 1923, 'early' as you claim, then let's assume she was around 30-40 years old when she died. An early death in those days would have been around that age, any older than 40 was beginning to be old and wouldn't have stood out as remarkable.
She would have been born between 1883 and 1893 then. Her son was already born, so he must have been born somewhere between 1901-1911 if she was 18, anywhere up to the year of her death in 1923. Which would make him between 84 and 106 years old. Stop me if I'm under or overestimating. So if your father is between 84 and 106, assuming you were born after he was 18 and before he was, say, 40, that would make you born somewhere between 1919 - 1951. I think. Maths ain't my strong point, but perhaps someone else can confirm my numbers. So you're probably between 88 and 56 years old. Except you have a newborn, so you can't be.
However, you say you made a promise to your father 34 years ago. To have made such a promise and still remember it, I will say you would have had to have been at least eight years old at the time, and that's being really, really over-generous. Pre-teen memory is very unreliable. However, let's go with that. So 34 years ago plus 8 could make you as young as 42, if by some chance your grandmother didn't die particularly young after all, or/and she had your father later in life and he had you later in life.
So let's say you're 42. Well, it's conceivable (pun intended) that you have a newborn, but not very likely. Of course, your story could have taken place many years ago, but in that case your memory of it is likely to be unreliable and therefore as good as useless. Time plus emotion makes liars of us all. Also it doesn't make much sense as your '34 years ago' bit is in the present tense.
I've made some perhaps unfair assumptions there, so let's have a look at what else your testimony gives us. Hmm, this is interesting:
The next time my husband returned, I pointed to a grave over by a tree and some distance over for him to check. When he got to that grave he stood and starred down for awhile, then slowly turned back toward the car and looked at me with the strangest look.
If the grave was some distance away and it was pouring with rain as you claim, how did you see the look on his face? I'm going to assume he had an umbrella rather than standing there with his mouth wide open in shock while it slowly filled up with water and drowned him. Right place for a quick burial though.
Equally, given the same conditions as above, how did he know which grave you were pointing at? If the grave was alone and isolated, perhaps an unusual colour or something, wouldn't that be a likely explanation for it grabbing your attention?
That's assuming any of the story is true, which it isn't.