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This is a good story that flows well at times, then seems to either bog down, or speed along at a rapid pace. The ending in particular, seems frantic and not well thought out. Some of your descriptions are a little vague, and in some cases, inaccurate (seawater temperature of 31 degrees for example).
Liz seems to be consumed with learning more about her ancestor Elspeth Coit. It's not clear why. You describe Elspeth as Liz's great grandmother, but shouldn't there be a couple more greats in that statement? There are 200 years between Elspeth and Liz, if you assume 50 years per generation, that's 4 generations, so she would be Liz's great great great grandmother. That's a minor point I know, but needs to be clarified.
My specific comments are below
Title: It fits the story well
Description: What made Elspeth famous? Liz discovers her diary, but there's very little said about the contents, other than the first page she reads that tells us why she was famous.
1. Furniture, and trunks filled with sketches, documents, and paintings lined the walls. I think you can remove the comma after Furniture. Also, describe the furniture for us? Perhaps something like this? Dusty furniture from a bygone age more suited to colonial times than the present, and trunks filled with sketches, documents, and paintings, lined the walls. My thought is, give the furniture in the attic a sense of age.
2. Very minor comment here. "Grotto. A small cave, or an artificial one in a park or garden, or an indoor structure, resembling a cave. Oh, great! Even if I find one, I have no idea what I'm looking for." You and I both know that her speaking the words she reads from the dictionary end with the word 'cave'. Yet with the quotation marks, it seems as if she's still reading from the dictionary. To highlight which part is the dictionary, and which is her commenting afterwards, use Italics to highlight the dictionary part.
3. A very general comment. You mention the location of the house as being on the tip of Jupiter Point in Setauket. I don't know if you meant this to be a fictional location or not, but there is a Setauket NY, but it's on Long Island sound, not facing the ocean. Also, there's no cliff overlooking anything there. Nothing raises very high above the ocean. At least that's how it seems. I could be wrong though. You might want to change the name to a fictional one is all. Just a thought.
4. As Liz approaches the ocean and heads to the west side, she starts down a sandy path. Then she slides into 'several of them'. Them what? We all know she's looking for grottos, but you never say she's entering grottos, just that she's entering them. You might want to make it obvious that's she is entering the grottos. When she decides to head for the east side, you write It took longer than expected since the west side was even rockier, and more dangerous. This is just a bit vague. You might want to add something about her traversing from the west side to the east, and as she went, it took longer than she expected. The west side was rockier than the east (she had gone to the west side first, so you might want to say that the east side was ever rockier and more dangerous.
5. Even in July, the ocean averaged a cool 31 degrees. We both know that 31 degrees is below freezing. But, with all the water action in the ocean, there's no way it will freeze, except maybe on the shore in low movement areas. But it's July, no way it will freeze. You should though, check the ocean temperatures online (I Googled it since the number 31 seemed way too low). https://seatemperature.info/july/sagaponack-water-... shows that the water temperature in Sagaponack NY in July averages around 65-70 degrees. I know you didn't state where Setauket is, but I would think that even in Maine the temperatures would be no where near 31 degrees. Change your temperature value to something in the high 50's at least.
6. There are at least two places in the paragraph immediately following her trek to the East side where Liz utters phrases in her head. An example is "I don't believe it!" Anytime she's thinking thoughts and not speaking them, use italics to separate that from actual speech.
7. After Liz find the metal box, things pick up pace. Her trek back to sandy ground seemed to go rather quickly, it was there she sat and read the first page of the 200 year old diary. You introduce Anna Smith, but who is she? What does she have to do with this story? I realize she's a patriot who seems to work alongside Elspeth, but what does that have to do with the story? One more thing here. She jumps up and runs toward the house. The trek down seemed to lengthy, if she's sitting in a 'rare area of sand' here, it's at the bottom of the cliffs. She wouldn't be able to run to the house, she has to climb the cliff first. IF she previously climbed the cliff, then she can't be sitting in sand.... Consistency, always consistency in your locations, characters, and descriptions.
8. One last thing. If Liz had found a diary that is 200 years old, it doesn't seem that reading it in the open with the wind blowing would be a good idea. It's been stored in a metal box, wrapped in cloth. The paper would be very fragile and crumble easily it seems. The diary would be best read under controlled conditions. She might give it to Uncle Rob, but I don't think having it read by anyone at the part would be a good idea.
Overall impressions: The story ends way too quickly, almost as if you were in a rush to complete it. Don't rush! Take your time and lead us there at the same pace that the rest of the story used. A good story that needs a bit of TLC attention to get the flow of it consistent.